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Reduced order optimization of large-scale nonlinear systems with nonlinear inequality constraints using steady state simulatorsPanagiotis Petsagkourakis?, Ioannis Bonis?, and Constantinos Theodoropoulos *?*Corresponding author. Tel.: +44 1612004386; fax: +44 1612367439. E-mail address: k.theodoropoulos@manchester.ac.uk (C. Theodoropoulos).?School of Chemical Engineering and Analytical Science, University of Manchester, Sackville St, Manchester M13 9PL, UKKeywords: Model reduction-based optimization, reduced Hessian, nonlinear inequality constraints, black-box simulator, large-scale optimization. AbstractTechnological advances have led to the widespread use of computational models of increasing complexity, in both industry and everyday life. This helps to improve the design, analysis and operation of complex systems. Many computational models in the field of engineering consist of systems of coupled nonlinear Partial Differential Equations (PDEs). As a result, optimization problems involving such models may lead to computational issues because of the large number of variables arising from the spatio-temporal discretization of the PDEs. In this work, we present a methodology for steady-state optimization, with non-linear inequality constraints of complex large-scale systems, for which only an input/output steady-state simulator is available. The proposed method is efficient for dissipative systems and is based on model reduction. This framework employs a two-step projection scheme followed by three different approaches for handling the nonlinear inequality constraints. In the first approach, partial reduction is implemented on the equality constraints, while the inequality constraints remain the same. In the second approach an aggregation function is applied in order to reduce the number of inequality constraints and solve the augmented problem. The final method applies slack variables to replace the one aggregated inequality from the previous method with an equality constraint without affecting the eigenspectrum of the system. Only low-order Jacobian and Hessian matrices are employed in the proposed formulations, utilizing only the available black-box simulator. The advantages and disadvantages of each approach are illustrated through the optimization of a tubular reactor where an exothermic reaction takes place. It is found that the approach involving the aggregation function can efficiently handle inequality constraints while significantly reducing the dimensionality of the system. 1. IntroductionThe optimization of large-scale systems is not a trivial task and has received significant attention over the years. Advances in technology have allowed the evolution of available simulators that can model accurately physical systems with high complexity, like COMSOLADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "id" : "ITEM-1", "issued" : { "date-parts" : [ [ "0" ] ] }, "title" : "COMSOL Multiphysics\u00ae v. 5.2. . 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However, in most cases, these simulators cannot perform optimization tasks. In addition, many of these simulators do not offer access to the underlying modelling equations. Distributed parameter systems (DPS)ADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1016/j.jprocont.2010.06.016", "ISBN" : "0959-1524", "ISSN" : "09591524", "abstract" : "Many industrial processes belong to distributed parameter systems (DPS) that have strong spatial-temporal dynamics. Modeling of DPS is difficult but essential to simulation, control and optimization. The first-principle modeling for known DPS often leads to the partial differential equation (PDE). Because it is an infinite-dimensional system, the model reduction (MR) is very necessary for real implementation. The model reduction often works with selection of basis functions (BF). Combination of different BF and MR results in different approaches. For unknown DPS, system identification is usually used to figure out unknown structure and parameters. Using various methods, different approaches are developed. Finally, a novel kernel-based approach is proposed for the complex DPS. 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In one model the linear dependence between electrolyte concentration and conductivity is accounted for, while in the other model it is not. A spectral element method is used to discretise the model equations and it is found that the error convergence rate with respect to the number of elements is faster compared to a finite difference method. The increased accuracy of the spectral element approach means that, for a similar level of solution accuracy, the model simulation computing time is approximately 50% of that of the finite difference method. This suggests that the spectral element model could be used for control and state estimation purposes. For a typical supercapacitor charging profile, the numerical solutions from both models closely match experimental voltage and current data. However, when the electrolyte is dilute or where there is a long charging time, a noticeable difference between the numerical solutions of the two models is observed. 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Unfortunately, the governing and constitutive equations of thermal processing models usually lead to complex sets of highly nonlinear partial differential equations (PDEs), which are difficult and costly to solve, especially in terms of computation time. We overcome such limitation by using a powerful model reduction technique based on proper orthogonal decomposition (POD) which yields simple, yet accurate, dynamic models still based on sound first principles. Model reduction is carried out by projecting the original set of PDEs on a low dimensional subspace which retains most of the relevant features of the original system. The resulting model consists of a small set of differential and algebraic equations (DAEs) suitable for real-time industrial applications (optimization and control). Further, this approach can be easily adapted to handle complex nonlinear convection-diffusion processes regardless of how irregular the domain geometry might be. \u00a9 2002 Elsevier Science Ltd. All rights reserved.", "author" : [ { "dropping-particle" : "", "family" : "Balsa-Canto", "given" : "Eva", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Alonso", "given" : "Antonio A.", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Banga", "given" : "Julio R.", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Journal of Food Engineering", "id" : "ITEM-1", "issue" : "3", "issued" : { "date-parts" : [ [ "2002" ] ] }, "page" : "227-234", "title" : "A novel, efficient and reliable method for thermal process design and optimization. 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The most common way to treat PDEs is to discretise them over a computational mesh producing large systems of nonlinear (dynamic) equations. 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Computing reduced versions of the (large-scale) system gradients can, therefore significantly enhance the applicability of deterministic optimization and control algorithms for large-scale systems. Moreover, when commercial simulators are employed, the systems’ gradients are not usually explicitly available to the user and need to be computed numerically. Automatic differentiationADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1007/978-3-642-30023-3_27", "ISBN" : "978-3-540-68935-5", "ISSN" : "1439-7358", "abstract" : "We present CasADi, a free, open-source software tool for fast, yet efficient\nsolution of nonlinear optimization problems in general and dynamic optimization problems in\nparticular. To the developer of algorithms for numerical optimization and to the advanced user of\nsuch algorithms, it offers a level of abstraction which is notably lower, and hence more flexible,\nthan that of algebraic modeling languages such as AMPL or GAMS, but higher than working with a\nconventional automatic differentiation (AD) tool.CasADi is best described as a minimalistic computer\nalgebra system (CAS) implementing automatic differentiation in eight different flavors. Similar to\nalgebraic modeling languages, it includes high-level interfaces to state-of-the-art numerical codes\nfor nonlinear programming, quadratic programming and integration of differential-algebraic\nequations. CasADi is implemented in self-contained C++ code and contains full-featured front-ends to\nPython and Octave for rapid prototyping. 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It is designed for problems with few degrees of freedom and is motivated by the need to use sparse matrix factorizations. The algorithm incorporates a correction vector that approximates the cross term Z(T)WYp(Y) in order to estimate the curvature in both the range and null spaces of the constraints. The algorithm can be considered to be, in some sense, a practical implementation of an algorithm of Coleman and Conn. 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Such optimization algorithms are often applied on process models developed within equation oriented process simulators. By partitioning the search direction into tangential and normal steps, these methods can exploit the structure of the equality constraints and adjust the remaining degrees of freedom in a lower dimensional space. Moreover, as shown in previous work, the barrier approach extended with a novel filter linear search algorithm has global and fast local convergence properties. However, convergence properties of the reduced-space barrier algorithm require regularity assumptions. In particular, the method may fail in the presence of linearly dependent active constraints. To deal with these questions, we modify the reduced-space barrier method in two ways. First, as the filter line search requires a feasibility restoration step, we develop and analyze an improved algorithm for this step, which is tailored to the reduced-space method. In addition, a dimension change procedure is proposed to address decomposition of problems with linearly dependent constraints. Finally, both approaches are implemented within a reduced-space version of IPOPT and numerical tests demonstrate the performance of the proposed modifications. ?? 2010 Elsevier Ltd.", "author" : [ { "dropping-particle" : "", "family" : "Wang", "given" : "Kexin", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Shao", "given" : "Zhijiang", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Biegler", "given" : "Lorenz T.", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Lang", "given" : "Yidong", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Qian", "given" : "Jixin", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Computers and Chemical Engineering", "id" : "ITEM-3", "issue" : "10", "issued" : { "date-parts" : [ [ "2011" ] ] }, "page" : "1994-2004", "publisher" : "Elsevier Ltd", "title" : "Robust extensions for reduced-space barrier NLP algorithms", "type" : "article-journal", "volume" : "35" }, "uris" : [ "" ] }, { "id" : "ITEM-4", "itemData" : { "DOI" : "10.1007/978-3-540-72699-9_13", "ISBN" : "978-3-540-72699-9", "abstract" : "The paper reports on recent progress in the real-time computation of constrained closed-loop optimal control, in particular the special case of nonlinear model predictive control, of large di.erential algebraic equations (DAE) systems arising e.g. from a MoL discretization of instationary PDE. Through a combination of a direct multiple shooting approach and an initial value embedding, a so-called ``real-time iteration'' approach has been developed in the last few years. One of the basic features is that in each iteration of the optimization process, new process data are being used. Through precomputation -- as far as possible -- of Hessian, gradients and QP factorizations the response time to perturbations of states and system parameters is minimized. We present and discuss new real-time algorithms for fast feasibility and optimality improvement that do not need to evaluate Jacobians online.", "author" : [ { "dropping-particle" : "", "family" : "Bock", "given" : "Hans Georg", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Diehl", "given" : "Moritz", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "K\u00fchl", "given" : "Peter", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Kostina", "given" : "Ekaterina", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Schi\u00f6der", "given" : "Johannes P", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Wirsching", "given" : "Leonard", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Assessment and Future Directions of Nonlinear Model Predictive Control", "editor" : [ { "dropping-particle" : "", "family" : "Findeisen", "given" : "Rolf", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Allg\u00f6wer", "given" : "Frank", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Biegler", "given" : "Lorenz T", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "id" : "ITEM-4", "issued" : { "date-parts" : [ [ "2007" ] ] }, "page" : "163-179", "publisher" : "Springer Berlin Heidelberg", "publisher-place" : "Berlin, Heidelberg", "title" : "Numerical Methods for Efficient and Fast Nonlinear Model Predictive Control", "type" : "chapter" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "^12\u201315", "plainTextFormattedCitation" : "12\u201315", "previouslyFormattedCitation" : "^12\u201315" }, "properties" : { }, "schema" : "" }12–15 have been developed based on sequential quadratic programming (SQP) and can be used for large-scale DPS with relatively few degrees of freedom. The advantage of these algorithms is that a low-order projection of the Hessian matrix is used, so less computational effort is required. The main idea is that suitable bases are constructed and used to project the large-scale system onto the low-dimensional subspace of the system’s independent variables, thus effectively reducing the dimensionality of the original system. However, these methods still require the construction (and inversion) of large-scale Jacobians and Hessians, hence requiring significant computational effort. To side-step these issues, rSQP methods have been combined with equation-free model reduction methodologiesADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1073/pnas.97.18.9840", "ISBN" : "0521899435", "ISSN" : "0027-8424", "PMID" : "10963656", "abstract" : "Evolutionary, pattern forming partial differential equations (PDEs) are often derived as limiting descriptions of microscopic, kinetic theory-based models of molecular processes (e.g., reaction and diffusion). The PDE dynamic behavior can be probed through direct simulation (time integration) or, more systematically, through stability/bifurcation calculations; time-stepper-based approaches, like the Recursive Projection Method [Shroff, G. M. & Keller, H. B. (1993) SIAM J. Numer. Anal. 30, 1099-1120] provide an attractive framework for the latter. We demonstrate an adaptation of this approach that allows for a direct, effective (\"coarse\") bifurcation analysis of microscopic, kinetic-based models; this is illustrated through a comparative study of the FitzHugh-Nagumo PDE and of a corresponding Lattice-Boltzmann model.", "author" : [ { "dropping-particle" : "", "family" : "Theodoropoulos", "given" : "C.", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Qian", "given" : "Y.-H.", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Kevrekidis", "given" : "I. 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In engineering practice, there are many limitations, either physical or technical, such as bounds of the system, of (dependent and independent) variables, and of properties. In addition, there are economic limitations, as well as limitations due to safety considerations (e.g. temperature bounds in the case of exothermic reactions where sudden temperature rise can lead to runaways). There are two main approaches for handling inequality constraints within the SQP frameworkADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1007/978-0-387-40065-5", "ISBN" : "1431-8598", "ISSN" : "10969101", "PMID" : "661941", "abstract" : "Summary: Numerical Optimization presents a comprehensive and up-to-date description of the most effective methods in continuous optimization. It responds to the growing interest in optimization in engineering, science, and business by focusing on the methods that are best suited to practical problems. For this new edition the book has been thoroughly updated throughout. There are new chapters on nonlinear interior methods and derivative-free methods for optimization, both of which are used widely in practice and the focus of much current research. Because of the emphasis on practical methods, as well as the extensive illustrations and exercises, the book is accessible to a wide audience. It can be used as a graduate text in engineering, operations research, mathematics, computer science, and business. It also serves as a handbook for researchers and practitioners in the field. The authors have strived to produce a text that is pleasant to read, informative, and rigorous - one that reveals both the beautiful nature of the discipline and its practical side.", "author" : [ { "dropping-particle" : "", "family" : "Nocedal", "given" : "Jorge", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Wright", "given" : "Stephen J", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Springer Series in Operations Research and Financial Engineering,", "id" : "ITEM-1", "issued" : { "date-parts" : [ [ "2006" ] ] }, "number-of-pages" : "XXII, 664 s. 85 illus.", "title" : "Numerical Optimization", "type" : "book" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "¹⁷", "plainTextFormattedCitation" : "17", "previouslyFormattedCitation" : "¹⁷" }, "properties" : { }, "schema" : "" }17: The first approach involves the sequential solution of inequality‐constrained QP sub-problems (IQP) and the second involves the sequential solution of equality constrained ones (EQP)ADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1007/978-0-387-40065-5", "ISBN" : "1431-8598", "ISSN" : "10969101", "PMID" : "661941", "abstract" : "Summary: Numerical Optimization presents a comprehensive and up-to-date description of the most effective methods in continuous optimization. It responds to the growing interest in optimization in engineering, science, and business by focusing on the methods that are best suited to practical problems. For this new edition the book has been thoroughly updated throughout. There are new chapters on nonlinear interior methods and derivative-free methods for optimization, both of which are used widely in practice and the focus of much current research. Because of the emphasis on practical methods, as well as the extensive illustrations and exercises, the book is accessible to a wide audience. It can be used as a graduate text in engineering, operations research, mathematics, computer science, and business. It also serves as a handbook for researchers and practitioners in the field. The authors have strived to produce a text that is pleasant to read, informative, and rigorous - one that reveals both the beautiful nature of the discipline and its practical side.", "author" : [ { "dropping-particle" : "", "family" : "Nocedal", "given" : "Jorge", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Wright", "given" : "Stephen J", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Springer Series in Operations Research and Financial Engineering,", "id" : "ITEM-1", "issued" : { "date-parts" : [ [ "2006" ] ] }, "number-of-pages" : "XXII, 664 s. 85 illus.", "title" : "Numerical Optimization", "type" : "book" }, "uris" : [ "" ] }, { "id" : "ITEM-2", "itemData" : { "DOI" : "10.1137/S1052623497325107", "ISBN" : "10.1137/S1052623497325107", "ISSN" : "1052-6234", "abstract" : "The design and implementation of a new algorithm for solving large nonlinear programming problems is described. It follows a barrier approach that employs sequential quadratic programming and trust regions to solve the subproblems occurring in the iteration. Both primal and primal-dual versions of the algorithm are developed, and their performance is illustrated in a set of numerical tests.", "author" : [ { "dropping-particle" : "", "family" : "Byrd", "given" : "R. 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Following the IQP rationale, at every iteration of the SQP (termed outer or major iteration) the nonlinear inequality constraints are linearized and included in the QP sub-problem, which in turn is solved using an active set approach. Conversely, in the EQP formulation, at every major iteration an estimation of the active subset of the inequality constraints is identified using estimates of the Lagrange multipliers and passed on to the QP as a working‐set, which leads to only equality‐constrained QP. This method has the advantage of lower computational cost and the utilization of simpler algorithms for quadratic programming. Both approaches have advantages and disadvantages, however none of the two can effectively handle the nonlinear inequality and equality constraints produced by a large-scale black box simulator for solution at real-time, as they require the full system gradients. In the framework of rSQP, the inequality constraints cannot be introduced directly as is the case in IQP. As a result SchulzADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "author" : [ { "dropping-particle" : "", "family" : "Schulz", "given" : "Volker H", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "id" : "ITEM-1", "issue" : "November 1995", "issued" : { "date-parts" : [ [ "1996" ] ] }, "title" : "Reduced SQP Methods for Large-Scale Optimal Control Problems in DAE with Application to Path Planning Problems for Satellite Mounted Robots", "type" : "article-journal" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "¹⁹", "plainTextFormattedCitation" : "19", "previouslyFormattedCitation" : "¹⁹" }, "properties" : { }, "schema" : "" }19 proposed a variant of rSQP, the so-called Partially Reduced SQP (PRSQP), which combines the strong properties of SQP and rSQP. The main idea of this approach is to exploit the structure of the null space of the equality constraints (or some of them) and handle the inequality constraints as in the SQP method. One widely-used approach to handle inequality constraints is the use of an aggregation function like the Kreisselmeier-Stainhauser (KS)ADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "author" : [ { "dropping-particle" : "", "family" : "Poon", "given" : "Nmk", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Martins", "given" : "J", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "2005 Canadian Aeronautics and Space Institute Annual General Meeting", "id" : "ITEM-1", "issued" : { "date-parts" : [ [ "2005" ] ] }, "page" : "1-12", "title" : "Adaptive Constraint Aggregation for Structural Optimization Using Adjoint Sensitivities", "type" : "article-journal" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "²⁰", "plainTextFormattedCitation" : "20", "previouslyFormattedCitation" : "²⁰" }, "properties" : { }, "schema" : "" }20. This method can reduce the number of inequality constraints to just one inequality and it can be combined with SQPADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "author" : [ { "dropping-particle" : "", "family" : "Poon", "given" : "Nmk", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Martins", "given" : "J", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "2005 Canadian Aeronautics and Space Institute Annual General Meeting", "id" : "ITEM-1", "issued" : { "date-parts" : [ [ "2005" ] ] }, "page" : "1-12", "title" : "Adaptive Constraint Aggregation for Structural Optimization Using Adjoint Sensitivities", "type" : "article-journal" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "²⁰", "plainTextFormattedCitation" : "20", "previouslyFormattedCitation" : "²⁰" }, "properties" : { }, "schema" : "" }20. The KS function has been also used as a barrier functionADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1016/S0098-1354(00)00591-3", "ISSN" : "00981354", "abstract" : "Process flexibility and design under uncertainty have been researched extensively in the literature. Problem formulations for flexibility include nested optimization problems and these can often be refined by substituting the optimality conditions for these nested problems. However, these reformulations are highly constrained and can be expensive to solve. In this paper we extend algorithms to solve these reformulated NLP problem under uncertainty by introducing two contributions to this approach. These are the use of a Constraint Aggregation function (KS function) and Smoothing Functions. We begin with basic properties of KS function. Next, we review a class of parametric smooth functions, used to replace the complementarity conditions of the KKT conditions with a well-behaved, smoothed nonlinear equality constraint. In this paper we apply the previous strategies to two specific problems: i) the?worst case algorithm?, that assesses design under uncertainty and, ii) the flexibility index and feasibility test formulations. In the first case, two new algorithms are derived, one of them being single level optimization problem. Next using similar ideas, both flexibility index and feasibility test are reformulated leading to a single non linear programming problem instead of a mixed integer non linear programming one. The new formulations are demonstrated on five different example problems where a CPU time reduction of more than 70 and 80% is demonstrated.", "author" : [ { "dropping-particle" : "", "family" : "Raspanti", "given" : "C G", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Bandoni", "given" : "J a", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Biegler", "given" : "L T", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Computers & Chemical Engineering", "id" : "ITEM-1", "issue" : "9-10", "issued" : { "date-parts" : [ [ "2000" ] ] }, "page" : "2193-2209", "title" : "New strategies for flexibility analysis and design under uncertainty", "type" : "article-journal", "volume" : "24" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "²¹", "plainTextFormattedCitation" : "21", "previouslyFormattedCitation" : "²¹" }, "properties" : { }, "schema" : "" }21 in chemical vapor deposition (CVD) applications in order to remove the nonlinear inequality constraints use them as a penalty term in the objective function through the KS function. Another interesting technique is the use of slack variablesADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1007/b106451", "ISBN" : "9780387249759", "author" : [ { "dropping-particle" : "", "family" : "Sun", "given" : "W", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Yuan", "given" : "Yx", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "id" : "ITEM-1", "issued" : { "date-parts" : [ [ "2006" ] ] }, "title" : "Optimization theory and methods: nonlinear programming", "type" : "article-journal" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "²²", "plainTextFormattedCitation" : "22", "previouslyFormattedCitation" : "²²" }, "properties" : { }, "schema" : "" }22, to turn inequality constraints into equalities, taking into account only the active inequality constraints. This method has some disadvantages as the number of the equality constraints may change from iteration to iteration. Therefore rSQP may fail to produce feasible solutions. Another interesting approach to solve the optimization problem with nonlinear inequality constraints is the barrier method, also known as interior point methodADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1007/b106451", "ISBN" : "9780387249759", "author" : [ { "dropping-particle" : "", "family" : "Sun", "given" : "W", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Yuan", "given" : "Yx", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "id" : "ITEM-1", "issued" : { "date-parts" : [ [ "2006" ] ] }, "title" : "Optimization theory and methods: nonlinear programming", "type" : "article-journal" }, "uris" : [ "" ] }, { "id" : "ITEM-2", "itemData" : { "author" : [ { "dropping-particle" : "", "family" : "Bazaraa", "given" : "", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Sherali", "given" : "", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Shetty", "given" : "", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "id" : "ITEM-2", "issued" : { "date-parts" : [ [ "2006" ] ] }, "title" : "Nonlinear programming Theory and Algorithms", "type" : "book" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "^22,23", "plainTextFormattedCitation" : "22,23", "previouslyFormattedCitation" : "^22,23" }, "properties" : { }, "schema" : "" }22,23, which solves an unconstrained problem by introducing a barrier function in the objective function. The most common barrier function is the logarithmic function. Thus, when one inequality is close to zero, the value of the barrier function increases exponentially. Hence, this method may produce sub-optimal results, as the inequality constraints can never be active. However, techniques exist that help the convergence of the algorithm to an active inequality taking advantage of central path methodologiesADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "author" : [ { "dropping-particle" : "", "family" : "Wills", "given" : "Adrian G", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Heath", "given" : "William P", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "id" : "ITEM-1", "issued" : { "date-parts" : [ [ "2007" ] ] }, "page" : "207-216", "title" : "Interior-Point Algorithms for Nonlinear Model Predictive Control", "type" : "article-journal" }, "uris" : [ "" ] }, { "id" : "ITEM-2", "itemData" : { "DOI" : "10.1109/TCST.2009.2017934", "ISBN" : "9550061051", "ISSN" : "0952-3499", "PMID" : "18494040", "abstract" : "A widely recognized shortcoming of model predictive control (MPC) is that it can usually only be used in applications with slow dynamics, where the sample time is measured in seconds or minutes. A well-known technique for implementing fast MPC is to compute the entire control law offline, in which case the online controller can be implemented as a lookup table. This method works well for systems with small state and input dimensions (say, no more than five), few constraints, and short time horizons. In this paper, we describe a collection of methods for improving the speed of MPC, using online optimization. These custom methods, which exploit the particular structure of the MPC problem, can compute the control action on the order of 100 times faster than a method that uses a generic optimizer. As an example, our method computes the control actions for a problem with 12 states, 3 controls, and horizon of 30 time steps (which entails solving a quadratic program with 450 variables and 1284 constraints) in around 5 ms, allowing MPC to be carried out at 200 Hz.", "author" : [ { "dropping-particle" : "", "family" : "Wang", "given" : "Yang", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Boyd", "given" : "Stephen", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Control Systems Technology, IEEE Transactions on", "id" : "ITEM-2", "issue" : "2", "issued" : { "date-parts" : [ [ "2010" ] ] }, "page" : "267-278", "title" : "Fast Model Predictive Control Using Online Optimization", "type" : "article-journal", "volume" : "18" }, "uris" : [ "" ] }, { "id" : "ITEM-3", "itemData" : { "ISBN" : "9780521833783", "author" : [ { "dropping-particle" : "", "family" : "Boyd", "given" : "S P", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Vandenberghe", "given" : "L", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "collection-title" : "Berichte \u00fcber verteilte messysteme", "id" : "ITEM-3", "issued" : { "date-parts" : [ [ "2004" ] ] }, "publisher" : "Cambridge University Press", "title" : "Convex Optimization", "type" : "book" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "^24\u201326", "plainTextFormattedCitation" : "24\u201326", "previouslyFormattedCitation" : "^24\u201326" }, "properties" : { }, "schema" : "" }24–26. A new optimisation technique was recently presented for large-scale dissipative systemsADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1016/j.ces.2011.09.033", "ISBN" : "0009-2509", "ISSN" : "00092509", "abstract" : "Most engineering systems can be accurately simulated using models consisting of Partial Differential Equations. Thus the challenging problem of PDE-constrained optimization arises naturally in engineering design. Issues surface due to the high number of variables involved and the use of specialized software for simulation which may not include an optimization option. In this work we present a methodology for the steady-state optimization of systems for which an input/output steady-state simulator is available. The proposed method is efficient for dissipative systems and is based on model reduction. This framework employs a two-step projection scheme, first onto the low-dimensional, adaptively computed, dominant subspace of the system and second onto the subspace of independent variables. Hence only low order Jacobian and Hessian matrices are used in this formulation, computed efficiently with directional perturbations. \u00a9 2011 Elsevier Ltd.", "author" : [ { "dropping-particle" : "", "family" : "Bonis", "given" : "Ioannis", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Theodoropoulos", "given" : "Constantinos", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Chemical Engineering Science", "id" : "ITEM-1", "issue" : "1", "issued" : { "date-parts" : [ [ "2012" ] ] }, "page" : "69-80", "publisher" : "Elsevier", "title" : "Model reduction-based optimization using large-scale steady-state simulators", "type" : "article-journal", "volume" : "69" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "²⁷", "plainTextFormattedCitation" : "27", "previouslyFormattedCitation" : "²⁷" }, "properties" : { }, "schema" : "" }27 based on equation-free methodsADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "author" : [ { "dropping-particle" : "", "family" : "Theodoropoulos", "given" : "Constantinos", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Qian", "given" : "Yue-hong", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Kevrekidis", "given" : "Ioannis G", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "id" : "ITEM-1", "issue" : "18", "issued" : { "date-parts" : [ [ "2000" ] ] }, "page" : "9840-9843", "title" : "\u201c Coarse \u201d stability and bifurcation analysis using time-steppers : A reaction-diffusion example", "type" : "article-journal", "volume" : "97" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "²⁸", "plainTextFormattedCitation" : "28", "previouslyFormattedCitation" : "²⁸" }, "properties" : { }, "schema" : "" }28, which exploits their dissipative nature for model order reduction (MOR). Dissipativity is expressed as separation of eigenvalues in the spectrum of the linearized system and therefore as a separation of system modes (or scales) to slow and fast onesADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1002/rnc.865", "ISSN" : "10498923", "abstract" : "We present an equation-free multiscale computational framework for the design of 'coarse' controllers for complex spatially distributed processes described by microscopic/mesoscopic evolution rules. We illustrate this framework by designing discrete-time, coarse linear controllers for a Lattice-Boltzmann (LB) scheme modelling a reaction-diffusion process (a kinetic-theory based realization of the FitzHugh-Nagumo equation dynamics in one spatial dimension). Short 'bursts' of appropriately initialized simulation of the LB model are used to extract the stationary states (stable and unstable) and to estimate the information required to design the coarse controller (e.g. the action of the coarse slow Jacobian of the process). Copyright (C) 2004 John Wiley Sons, Ltd.", "author" : [ { "dropping-particle" : "", "family" : "Armaou", "given" : "Antonios", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Siettos", "given" : "Constantinos I.", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Kevrekidis", "given" : "Ioannis G.", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "International Journal of Robust and Nonlinear Control", "id" : "ITEM-1", "issue" : "2", "issued" : { "date-parts" : [ [ "2004" ] ] }, "page" : "89-111", "title" : "Time-steppers and 'coarse' control of distributed microscopic processes", "type" : "article-journal", "volume" : "14" }, "uris" : [ "" ] }, { "id" : "ITEM-2", "itemData" : { "DOI" : "10.1137/0730057", "ISBN" : "00361429", "ISSN" : "0036-1429", "PMID" : "17746741", "abstract" : "Fixed-point iterative procedures for solving nonlinear parameter dependent problems can converge for some interval of parameter values and diverge as the parameter changes. The Recursive Projection Method (RPM), which stabilizes such procedures by computing a projection onto the unstable subspace is presented. On this subspace a Newton or special Newton iteration is performed, and the fixed-point iteration is used on the complement. As continuation in the parameter proceeds, the projection is efficiently updated, possibly increasing or decreasing the dimension of the unstable subspace. The method is extremely effective when the dimension of the unstable subspace is small compared to the dimension of the system. Convergence proofs are given and pseudo-arclength continuation on the unstable subspace is introduced to allow continuation past folds. Examples are presented for an important application of the RPM in which a \u201cblack-box\u201d time integration scheme is stabilized, enabling it to compute unstable steady...", "author" : [ { "dropping-particle" : "", "family" : "Shroff", "given" : "Gautam M.", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Keller", "given" : "Herbert B.", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "SIAM Journal on Numerical Analysis", "id" : "ITEM-2", "issue" : "4", "issued" : { "date-parts" : [ [ "1993" ] ] }, "page" : "1099-1120", "title" : "Stabilization of Unstable Procedures: The Recursive Projection Method", "type" : "article-journal", "volume" : "30" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "^29,30", "plainTextFormattedCitation" : "29,30", "previouslyFormattedCitation" : "^29,30" }, "properties" : { }, "schema" : "" }29,30. This separation has been used in various ways within the MOR context, leading to different formulationsADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "author" : [ { "dropping-particle" : "", "family" : "Theodoropoulos", "given" : "C", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Lecture Notes in Computational Science and Engineering Vol 75", "id" : "ITEM-1", "issued" : { "date-parts" : [ [ "2011" ] ] }, "page" : "37-61", "title" : "Optimisation and linear control of large scale nonlinear systems: A review and a suite of model reduction-based techniques", "type" : "chapter" }, "uris" : [ "" ] }, { "id" : "ITEM-2", "itemData" : { "DOI" : "Doi 10.1137/040604716", "ISBN" : "1540-3459", "ISSN" : "15403459", "abstract" : "A reduced model based optimization strategy is presented for the cases where input/ output codes are the process simulators of choice, and thus system Jacobians and even system equations are not explicitly available to the user. The former is the case when commercial software packages or legacy codes are used to simulate a large-scale system and the latter when microscopic or multiscale simulators are employed. When such black-box dynamic simulators are used, we perform optimization by combining the recursive projection method [ G. M. Shroff and H. B. Keller, SIAM J. Numer. Anal., 30 ( 1993), pp. 1099 - 1120] which identifies the ( typically) low-dimensional slow dynamics of the (dissipative) model with a second reduction to the low-dimensional subspace of the decision variables. This results in the solution of a low-order unconstrained optimization problem. Optimal conditions are then computed in an efficient way using only low-dimensional numerical approximations of gradients and Hessians. The tubular reactor is used as an illustrative example to demonstrate this model reduction-based optimization methodology.", "author" : [ { "dropping-particle" : "", "family" : "Luna-Ortiz", "given" : "E", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Theodoropoulos", "given" : "Constantinos", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Multiscale Modeling & Simulation", "id" : "ITEM-2", "issue" : "2", "issued" : { "date-parts" : [ [ "2005" ] ] }, "page" : "691-708", "title" : "An input/output model reduction-based optimization scheme for large-scale systems", "type" : "article-journal", "volume" : "4" }, "uris" : [ "" ] }, { "id" : "ITEM-3", "itemData" : { "author" : [ { "dropping-particle" : "", "family" : "Theodoropoulos", "given" : "C", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Luna-ortiz", "given" : "Eduardo", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena", "id" : "ITEM-3", "issued" : { "date-parts" : [ [ "2006" ] ] }, "page" : "535-560", "title" : "A Reduced Input / Output Dynamic Optimisation Method", "type" : "chapter" }, "uris" : [ "" ] }, { "id" : "ITEM-4", "itemData" : { "DOI" : "10.1016/j.ces.2011.09.033", "ISBN" : "0009-2509", "ISSN" : "00092509", "abstract" : "Most engineering systems can be accurately simulated using models consisting of Partial Differential Equations. Thus the challenging problem of PDE-constrained optimization arises naturally in engineering design. Issues surface due to the high number of variables involved and the use of specialized software for simulation which may not include an optimization option. In this work we present a methodology for the steady-state optimization of systems for which an input/output steady-state simulator is available. The proposed method is efficient for dissipative systems and is based on model reduction. This framework employs a two-step projection scheme, first onto the low-dimensional, adaptively computed, dominant subspace of the system and second onto the subspace of independent variables. Hence only low order Jacobian and Hessian matrices are used in this formulation, computed efficiently with directional perturbations. \u00a9 2011 Elsevier Ltd.", "author" : [ { "dropping-particle" : "", "family" : "Bonis", "given" : "Ioannis", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Theodoropoulos", "given" : "Constantinos", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Chemical Engineering Science", "id" : "ITEM-4", "issue" : "1", "issued" : { "date-parts" : [ [ "2012" ] ] }, "page" : "69-80", "publisher" : "Elsevier", "title" : "Model reduction-based optimization using large-scale steady-state simulators", "type" : "article-journal", "volume" : "69" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "^{27,31\u201333}", "plainTextFormattedCitation" : "27,31\u201333", "previouslyFormattedCitation" : "^{27,31\u201333}" }, "properties" : { }, "schema" : "" }27,31–33. Nevertheless, none of the above MOR-based methods has dealt with problems that include nonlinear constraints.In this paper, the aforementioned model reduction techniqueADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1016/j.ces.2011.09.033", "ISBN" : "0009-2509", "ISSN" : "00092509", "abstract" : "Most engineering systems can be accurately simulated using models consisting of Partial Differential Equations. Thus the challenging problem of PDE-constrained optimization arises naturally in engineering design. Issues surface due to the high number of variables involved and the use of specialized software for simulation which may not include an optimization option. In this work we present a methodology for the steady-state optimization of systems for which an input/output steady-state simulator is available. The proposed method is efficient for dissipative systems and is based on model reduction. This framework employs a two-step projection scheme, first onto the low-dimensional, adaptively computed, dominant subspace of the system and second onto the subspace of independent variables. Hence only low order Jacobian and Hessian matrices are used in this formulation, computed efficiently with directional perturbations. \u00a9 2011 Elsevier Ltd.", "author" : [ { "dropping-particle" : "", "family" : "Bonis", "given" : "Ioannis", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Theodoropoulos", "given" : "Constantinos", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Chemical Engineering Science", "id" : "ITEM-1", "issue" : "1", "issued" : { "date-parts" : [ [ "2012" ] ] }, "page" : "69-80", "publisher" : "Elsevier", "title" : "Model reduction-based optimization using large-scale steady-state simulators", "type" : "article-journal", "volume" : "69" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "²⁷", "plainTextFormattedCitation" : "27", "previouslyFormattedCitation" : "²⁷" }, "properties" : { }, "schema" : "" }27 was exploited in conjunction with three different approaches to handle large-scale systems with nonlinear constraints. The first approach combines PRSQP with equation-free model reduction, to reduce the dimensionality of equality constraints only. The second approach adds the feature of constraint aggregation34, where all the inequality constraints are replaced by a single KS function. This way large-scale inequality constraints can be effectively handled. In the last approach, a slack variable is employed to turn the aggregated inequality into an additional equality constraint. Slack variables could potentially produce difficulties in the model reduction step; however we provide a proof that the aggregation of the inequalities helps to avoid such issues. The rest of the paper is organized as follows: Section 2 presents the background for this work including a brief overview of the Partial reduced Sequential Quadratic Programming method, the constraint aggregation method and the equation-free model reduction framework. In section 3 we present the new methodology developed in this work for handling large-scale nonlinear optimisation problems with nonlinear inequalities. The proposed schemes are discussed alongside with proofs of equivalence of the computed optima. We apply the 3 schemes developed to an illustrative case study, the optimization of a tubular reactor, in section 4. Finally, a comparison of the three schemes along with relevant conclusions is presented in section 5. 2. Background 2.1 Partial Reduced Sequential Quadratic ProgrammingPartial Reduced Sequential Quadratic Programming (PRSQP) was introducedADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "author" : [ { "dropping-particle" : "", "family" : "Schulz", "given" : "Volker H", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "id" : "ITEM-1", "issue" : "November 1995", "issued" : { "date-parts" : [ [ "1996" ] ] }, "title" : "Reduced SQP Methods for Large-Scale Optimal Control Problems in DAE with Application to Path Planning Problems for Satellite Mounted Robots", "type" : "article-journal" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "¹⁹", "plainTextFormattedCitation" : "19", "previouslyFormattedCitation" : "¹⁹" }, "properties" : { }, "schema" : "" }19 in order to extend reduced Hessian methods for problems with “additional” nonlinear equality and inequality constraints. PRSQP reduces the space of (some) equality constraints and the rest of the equality and inequality constraints are treated in a similar manner as in SQP. Thus, the extra constraints (both inequality and equality constraints) are passed to the QP sub-problem. The problem formulation is described as follows: minfx s.t. Gx=0hx≤0xL≤x≤xU (1)Here fx is the objective function, x∈RN+dof is the vector of (dependent (u) and independent (xdof)) variables. G:RN+dof→RN represents the N equality constraints, and h:RN+dof→RNin the Nin inequality constraints. Additionally, the Lagrange function, L, is defined as followsLx=fx+λΤGx+λinTh(x) (2)where the Lagrange multipliers, λ, that correspond to equality constraints and the ones corresponding to inequality constraints, λin, are computed directly from the solution of the QP sub-problem. As mentioned above, large-scale problems, i.e. systems with large number of equality and inequality constraints still require the construction (and inversion) of large full-scale Jacobians and Hessians, compromising the computational efficiency of the optimization method. For cases with large number of inequality constraints, aggregation methods can be applied.2.2. Constraints aggregation The KS function was first presented by G. Kreisselmeier and R. SteinhauserADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "author" : [ { "dropping-particle" : "", "family" : "Poon", "given" : "Nmk", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Martins", "given" : "J", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "2005 Canadian Aeronautics and Space Institute Annual General Meeting", "id" : "ITEM-1", "issued" : { "date-parts" : [ [ "2005" ] ] }, "page" : "1-12", "title" : "Adaptive Constraint Aggregation for Structural Optimization Using Adjoint Sensitivities", "type" : "article-journal" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "²⁰", "plainTextFormattedCitation" : "20", "previouslyFormattedCitation" : "²⁰" }, "properties" : { }, "schema" : "" }20. The function contains an ‘aggregation parameter’, ρ which is equivalent to the penalty factor in penalty methods. This formulation was first used to combine multiple objectives and constraints into a single function and has been utilised in a wide variety of applications such as CVD optimizationADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1016/S0098-1354(00)00591-3", "ISSN" : "00981354", "abstract" : "Process flexibility and design under uncertainty have been researched extensively in the literature. Problem formulations for flexibility include nested optimization problems and these can often be refined by substituting the optimality conditions for these nested problems. However, these reformulations are highly constrained and can be expensive to solve. In this paper we extend algorithms to solve these reformulated NLP problem under uncertainty by introducing two contributions to this approach. These are the use of a Constraint Aggregation function (KS function) and Smoothing Functions. We begin with basic properties of KS function. Next, we review a class of parametric smooth functions, used to replace the complementarity conditions of the KKT conditions with a well-behaved, smoothed nonlinear equality constraint. In this paper we apply the previous strategies to two specific problems: i) the?worst case algorithm?, that assesses design under uncertainty and, ii) the flexibility index and feasibility test formulations. In the first case, two new algorithms are derived, one of them being single level optimization problem. Next using similar ideas, both flexibility index and feasibility test are reformulated leading to a single non linear programming problem instead of a mixed integer non linear programming one. The new formulations are demonstrated on five different example problems where a CPU time reduction of more than 70 and 80% is demonstrated.", "author" : [ { "dropping-particle" : "", "family" : "Raspanti", "given" : "C G", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Bandoni", "given" : "J a", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Biegler", "given" : "L T", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Computers & Chemical Engineering", "id" : "ITEM-1", "issue" : "9-10", "issued" : { "date-parts" : [ [ "2000" ] ] }, "page" : "2193-2209", "title" : "New strategies for flexibility analysis and design under uncertainty", "type" : "article-journal", "volume" : "24" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "²¹", "plainTextFormattedCitation" : "21", "previouslyFormattedCitation" : "²¹" }, "properties" : { }, "schema" : "" }21 and structural optimizationADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "author" : [ { "dropping-particle" : "", "family" : "Poon", "given" : "Nmk", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Martins", "given" : "J", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "2005 Canadian Aeronautics and Space Institute Annual General Meeting", "id" : "ITEM-1", "issued" : { "date-parts" : [ [ "2005" ] ] }, "page" : "1-12", "title" : "Adaptive Constraint Aggregation for Structural Optimization Using Adjoint Sensitivities", "type" : "article-journal" }, "uris" : [ "" ] }, { "id" : "ITEM-2", "itemData" : { "abstract" : "This paper explores different methods of constraint aggregation for numerical optimization. The main motive is the aggregation of stress constraints in structural weight minimization problems in order to reduce the cost of adjoint sensitivity calculations and hence the overall cost of the optimization. We analyze existing approaches such as considering all constraints individually, taking the maximum of the constraints and using the Kreisselmeier\u2013Steinhauser function. A new adaptive approach based on the Kreisselmeier\u2013Steinhauser function is proposed and is shown to significantly increase the accuracy of the results when a large number of constraints is active at the optimum.", "author" : [ { "dropping-particle" : "", "family" : "Martins", "given" : "Joaquim R R a", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Poon", "given" : "Nicholas M K", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Proceedings of 6th World Congress on Structural and Multidisciplinary Optimization", "id" : "ITEM-2", "issue" : "June", "issued" : { "date-parts" : [ [ "2005" ] ] }, "page" : "1-10", "title" : "On Structural Optimization Using Constraint Aggregation", "type" : "article-journal" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "^20,34", "plainTextFormattedCitation" : "20,34", "previouslyFormattedCitation" : "^20,34" }, "properties" : { }, "schema" : "" }20,34. The KS function can be used to aggregate inequality constraints and is described as follows:KShi=1ρln?(i=1Ninexp?(ρhi)) (3)An equivalent expression is as follows:KShi=M+1ρln?(i=1Ninexp?(ρhi-M)) (4)where ρ and M are design parameters. The second expression provides better behaviour when one or more inequalities are positive, due to numerical difficulties that may be caused by the exponential term. The design parameter M is suggestedADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1016/S0098-1354(00)00591-3", "ISSN" : "00981354", "abstract" : "Process flexibility and design under uncertainty have been researched extensively in the literature. Problem formulations for flexibility include nested optimization problems and these can often be refined by substituting the optimality conditions for these nested problems. However, these reformulations are highly constrained and can be expensive to solve. In this paper we extend algorithms to solve these reformulated NLP problem under uncertainty by introducing two contributions to this approach. These are the use of a Constraint Aggregation function (KS function) and Smoothing Functions. We begin with basic properties of KS function. Next, we review a class of parametric smooth functions, used to replace the complementarity conditions of the KKT conditions with a well-behaved, smoothed nonlinear equality constraint. In this paper we apply the previous strategies to two specific problems: i) the?worst case algorithm?, that assesses design under uncertainty and, ii) the flexibility index and feasibility test formulations. In the first case, two new algorithms are derived, one of them being single level optimization problem. Next using similar ideas, both flexibility index and feasibility test are reformulated leading to a single non linear programming problem instead of a mixed integer non linear programming one. The new formulations are demonstrated on five different example problems where a CPU time reduction of more than 70 and 80% is demonstrated.", "author" : [ { "dropping-particle" : "", "family" : "Raspanti", "given" : "C G", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Bandoni", "given" : "J a", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Biegler", "given" : "L T", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Computers & Chemical Engineering", "id" : "ITEM-1", "issue" : "9-10", "issued" : { "date-parts" : [ [ "2000" ] ] }, "page" : "2193-2209", "title" : "New strategies for flexibility analysis and design under uncertainty", "type" : "article-journal", "volume" : "24" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "²¹", "plainTextFormattedCitation" : "21", "previouslyFormattedCitation" : "²¹" }, "properties" : { }, "schema" : "" }21 to be the maximum value of the inequality constraints. Properties of the KS function can be found in Raspanti et alADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1016/S0098-1354(00)00591-3", "ISSN" : "00981354", "abstract" : "Process flexibility and design under uncertainty have been researched extensively in the literature. Problem formulations for flexibility include nested optimization problems and these can often be refined by substituting the optimality conditions for these nested problems. However, these reformulations are highly constrained and can be expensive to solve. In this paper we extend algorithms to solve these reformulated NLP problem under uncertainty by introducing two contributions to this approach. These are the use of a Constraint Aggregation function (KS function) and Smoothing Functions. We begin with basic properties of KS function. Next, we review a class of parametric smooth functions, used to replace the complementarity conditions of the KKT conditions with a well-behaved, smoothed nonlinear equality constraint. In this paper we apply the previous strategies to two specific problems: i) the?worst case algorithm?, that assesses design under uncertainty and, ii) the flexibility index and feasibility test formulations. In the first case, two new algorithms are derived, one of them being single level optimization problem. Next using similar ideas, both flexibility index and feasibility test are reformulated leading to a single non linear programming problem instead of a mixed integer non linear programming one. 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Equation-Free Model ReductionEquation-free model reductionADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1073/pnas.97.18.9840", "ISBN" : "0521899435", "ISSN" : "0027-8424", "PMID" : "10963656", "abstract" : "Evolutionary, pattern forming partial differential equations (PDEs) are often derived as limiting descriptions of microscopic, kinetic theory-based models of molecular processes (e.g., reaction and diffusion). The PDE dynamic behavior can be probed through direct simulation (time integration) or, more systematically, through stability/bifurcation calculations; time-stepper-based approaches, like the Recursive Projection Method [Shroff, G. M. & Keller, H. B. (1993) SIAM J. Numer. Anal. 30, 1099-1120] provide an attractive framework for the latter. We demonstrate an adaptation of this approach that allows for a direct, effective (\"coarse\") bifurcation analysis of microscopic, kinetic-based models; this is illustrated through a comparative study of the FitzHugh-Nagumo PDE and of a corresponding Lattice-Boltzmann model.", "author" : [ { "dropping-particle" : "", "family" : "Theodoropoulos", "given" : "C.", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Qian", "given" : "Y.-H.", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Kevrekidis", "given" : "I. 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The system’s dissipativity can be expressed as a gap in the spectrum of the eigenvalues of the linearized problem (Jacobian of the equality constraints). A (usually) small number of eigenvalues is clustered near the imaginary axis (red dots in Figure 1). These eigenvalues correspond to the slow and/or unstable modes of the system. The rest of the eigenvalues beyond the gap (blue dots in Figure one) correspond to the fast modes. This idealized spectrum is illustrated in REF _Ref451182026 \h \* MERGEFORMAT Figure 1.Figure SEQ Figure \* ARABIC 1. Idealized separation of scales for the eigenvalues of a dissipative system.The eigenvalues affect the stability of the static statesADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1137/0730057", "ISBN" : "00361429", "ISSN" : "0036-1429", "PMID" : "17746741", "abstract" : "Fixed-point iterative procedures for solving nonlinear parameter dependent problems can converge for some interval of parameter values and diverge as the parameter changes. The Recursive Projection Method (RPM), which stabilizes such procedures by computing a projection onto the unstable subspace is presented. On this subspace a Newton or special Newton iteration is performed, and the fixed-point iteration is used on the complement. As continuation in the parameter proceeds, the projection is efficiently updated, possibly increasing or decreasing the dimension of the unstable subspace. The method is extremely effective when the dimension of the unstable subspace is small compared to the dimension of the system. Convergence proofs are given and pseudo-arclength continuation on the unstable subspace is introduced to allow continuation past folds. Examples are presented for an important application of the RPM in which a \u201cblack-box\u201d time integration scheme is stabilized, enabling it to compute unstable steady...", "author" : [ { "dropping-particle" : "", "family" : "Shroff", "given" : "Gautam M.", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Keller", "given" : "Herbert B.", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "SIAM Journal on Numerical Analysis", "id" : "ITEM-1", "issue" : "4", "issued" : { "date-parts" : [ [ "1993" ] ] }, "page" : "1099-1120", "title" : "Stabilization of Unstable Procedures: The Recursive Projection Method", "type" : "article-journal", "volume" : "30" }, "uris" : [ "" ] }, { "id" : "ITEM-2", "itemData" : { "author" : [ { "dropping-particle" : "", "family" : "Schulz", "given" : "Volker H", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "id" : "ITEM-2", "issue" : "November 1995", "issued" : { "date-parts" : [ [ "1996" ] ] }, "title" : "Reduced SQP Methods for Large-Scale Optimal Control Problems in DAE with Application to Path Planning Problems for Satellite Mounted Robots", "type" : "article-journal" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "^19,30", "plainTextFormattedCitation" : "19,30", "previouslyFormattedCitation" : "^19,30" }, "properties" : { }, "schema" : "" }19,30. In fact, the rightmost, slow modes in the idealized eigenspectrum ( REF _Ref451182026 \h \* MERGEFORMAT Figure 1) enslave the rest and determine the system’s stabilityADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1137/1.9781611970739", "ISBN" : "0719033861 (UK) 0470218207 (US)", "author" : [ { "dropping-particle" : "", "family" : "Saad", "given" : "Yousef", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Algorithms and architectures for advanced scientific computing", "id" : "ITEM-1", "issued" : { "date-parts" : [ [ "1992" ] ] }, "page" : "346 p.", "title" : "Numerical methods for large eigenvalue problems", "type" : "article-journal" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "³⁵", "plainTextFormattedCitation" : "35", "previouslyFormattedCitation" : "³⁵" }, "properties" : { }, "schema" : "" }35. The number, m, of the slow modes depends on the separation of scales whose existence has been proven for parabolic PDEsADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1016/pchemeng.2006.05.025", "ISBN" : "0098-1354", "ISSN" : "00981354", "abstract" : "In this work, we present an overview of recently developed methods for control and optimization of complex process systems described by multiscale models. We primarily discuss methods developed in the context of our previous research work and use examples of thin film growth processes to motivate the development of these methods and illustrate their application. ?? 2006 Elsevier Ltd. All rights reserved.", "author" : [ { "dropping-particle" : "", "family" : "Christofides", "given" : "Panagiotis D.", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Armaou", "given" : "Antonios", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Computers and Chemical Engineering", "id" : "ITEM-1", "issue" : "10-12", "issued" : { "date-parts" : [ [ "2006" ] ] }, "page" : "1670-1686", "title" : "Control and optimization of multiscale process systems", "type" : "article-journal", "volume" : "30" }, "uris" : [ "" ] }, { "id" : "ITEM-2", "itemData" : { "DOI" : "10.1016/S0005-1098(02)00304-7", "ISBN" : "0-7803-7061-9", "ISSN" : "00051098", "abstract" : "This paper develops a general framework for the analysis and control of parabolic partial differential equations (PDE) systems with input constraints. Initially, Galerkin's method is used for the derivation of ordinary differential equation (ODE) system that capture the dominant dynamics of the PDE system. This ODE systems are then used as the basis for the synthesis, via Lyapunov techniques, of stabilizing bounded nonlinear state and output feedback control laws that provide an explicit characterization of the sets of admissible initial conditions and admissible control actuator locations that can be used to guarantee closed-loop stability in the presence of constraints. Precise conditions that guarantee stability of the constrained closed-loop parabolic PDE system are provided in terms of the separation between the fast and slow eigenmodes of the spatial differential operator. The theoretical results are used to stabilize an unstable steady-state of a diffusion-reaction process using constrained control action. ?? 2002 Elsevier Science Ltd. All rights reserved.", "author" : [ { "dropping-particle" : "", "family" : "El-Farra", "given" : "Nael H.", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Armaou", "given" : "Antonios", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Christofides", "given" : "Panagiotis D.", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Automatica", "id" : "ITEM-2", "issue" : "4", "issued" : { "date-parts" : [ [ "2003" ] ] }, "page" : "715-725", "title" : "Analysis and control of parabolic PDE systems with input constraints", "type" : "article-journal", "volume" : "39" }, "uris" : [ "" ] }, { "id" : "ITEM-3", "itemData" : { "author" : [ { "dropping-particle" : "", "family" : "Friedman", "given" : "Anver", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "id" : "ITEM-3", "issued" : { "date-parts" : [ [ "1964" ] ] }, "publisher" : "Dover Publications INC", "publisher-place" : "Mineola, New York", "title" : "Partial differential equations of parabolic type", "type" : "book" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "^36\u201338", "plainTextFormattedCitation" : "36\u201338", "previouslyFormattedCitation" : "^36\u201338" }, "properties" : { }, "schema" : "" }36–38. A basis spanning the dominant subspace of the system, of size m, can efficiently be computed using subspace iterations or Krylov subspace-based algorithms like Arnoldi iterationsADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1137/1.9781611970739", "ISBN" : "0719033861 (UK) 0470218207 (US)", "author" : [ { "dropping-particle" : "", "family" : "Saad", "given" : "Yousef", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Algorithms and architectures for advanced scientific computing", "id" : "ITEM-1", "issued" : { "date-parts" : [ [ "1992" ] ] }, "page" : "346 p.", "title" : "Numerical methods for large eigenvalue problems", "type" : "article-journal" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "³⁵", "plainTextFormattedCitation" : "35", "previouslyFormattedCitation" : "³⁵" }, "properties" : { }, "schema" : "" }35. The size of the basis can be heuristically derived or adaptively computed as in the case of the recursive projection method (RPM)ADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1137/0730057", "ISBN" : "00361429", "ISSN" : "0036-1429", "PMID" : "17746741", "abstract" : "Fixed-point iterative procedures for solving nonlinear parameter dependent problems can converge for some interval of parameter values and diverge as the parameter changes. The Recursive Projection Method (RPM), which stabilizes such procedures by computing a projection onto the unstable subspace is presented. On this subspace a Newton or special Newton iteration is performed, and the fixed-point iteration is used on the complement. As continuation in the parameter proceeds, the projection is efficiently updated, possibly increasing or decreasing the dimension of the unstable subspace. The method is extremely effective when the dimension of the unstable subspace is small compared to the dimension of the system. Convergence proofs are given and pseudo-arclength continuation on the unstable subspace is introduced to allow continuation past folds. Examples are presented for an important application of the RPM in which a \u201cblack-box\u201d time integration scheme is stabilized, enabling it to compute unstable steady...", "author" : [ { "dropping-particle" : "", "family" : "Shroff", "given" : "Gautam M", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Keller", "given" : "Herbert B", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "SIAM Journal on Numerical Analysis", "id" : "ITEM-1", "issue" : "4", "issued" : { "date-parts" : [ [ "1993" ] ] }, "page" : "1099-1120", "title" : "Stabilization of Unstable Procedures: The Recursive Projection Method", "type" : "article-journal", "volume" : "30" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "³⁹", "plainTextFormattedCitation" : "39", "previouslyFormattedCitation" : "³⁹" }, "properties" : { }, "schema" : "" }39. These techniques follow the matrix-free concept as only evaluation of matrix-vector products is needed. As a result, even though the systems’ equations and/or Jacobians may not be explicitly available, the calculation of the basis is efficient and feasible. Assume that G=0 represents the system equations, as in eq. 1 above, contained within a (steady-state) black box simulator, G:RN+dof→RN being Lebesque integrable. If P is the dominant sub-space of the system and Q its orthogonal complement then:P⊕Q=RN (5)An orthonormal basis Z?RN×m for the subspace P and a projector P are defined asP=ZZT (6) ZTZ=I (7)As discussed above, an approximation, Z, of Z is computed through Krylov or Anrnoldi iterations. The vector of system states, u, can be replaced with its low-dimensional projection, υ, onto P: υ=ZTu. The reduced Jacobian, H?Rm×m, is then computed by the restriction of the full-scale Jacobian, J, onto Z:H=ZTJZ (8)side-stepping the need to calculate the full Jacobian. The reduced Jacobian is efficiently computed through m numerical perturbations for ε>0:JZj=12εGu+εZj-Gu-εΖj, j=1,…,m (9) This scheme follows the matrix-free concept, to reduce mainly memory requirements and has roots in the Recursive Projection MethodADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1137/0730057", "ISBN" : "00361429", "ISSN" : "0036-1429", "PMID" : "17746741", "abstract" : "Fixed-point iterative procedures for solving nonlinear parameter dependent problems can converge for some interval of parameter values and diverge as the parameter changes. The Recursive Projection Method (RPM), which stabilizes such procedures by computing a projection onto the unstable subspace is presented. On this subspace a Newton or special Newton iteration is performed, and the fixed-point iteration is used on the complement. As continuation in the parameter proceeds, the projection is efficiently updated, possibly increasing or decreasing the dimension of the unstable subspace. The method is extremely effective when the dimension of the unstable subspace is small compared to the dimension of the system. Convergence proofs are given and pseudo-arclength continuation on the unstable subspace is introduced to allow continuation past folds. Examples are presented for an important application of the RPM in which a \u201cblack-box\u201d time integration scheme is stabilized, enabling it to compute unstable steady...", "author" : [ { "dropping-particle" : "", "family" : "Shroff", "given" : "Gautam M.", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Keller", "given" : "Herbert B.", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "SIAM Journal on Numerical Analysis", "id" : "ITEM-1", "issue" : "4", "issued" : { "date-parts" : [ [ "1993" ] ] }, "page" : "1099-1120", "title" : "Stabilization of Unstable Procedures: The Recursive Projection Method", "type" : "article-journal", "volume" : "30" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "³⁰", "plainTextFormattedCitation" : "30", "previouslyFormattedCitation" : "³⁰" }, "properties" : { }, "schema" : "" }30. Multiplying ZTwith JZ produces the desired reduced JacobianADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1002/aic", "ISBN" : "9783902661548", "ISSN" : "14746670", "PMID" : "23641116", "abstract" : "The effect of fouling in heat-transfer devices (HTDs) is complicated by aging of the fouling deposits. Aging is, like deposition, often sensitive to temperature, so that heat transfer, deposition, and aging are coupled phenomena. Ishiyama et al. (AIChE J. 2010;56:531\u2013545) presented a distributed model of the aging of deposits formed by chemical reaction fouling and illustrated its effect on thermal and hydraulic performance of a HTD operating in the turbulent flow regime. Two-layer models, simpler than the distributed model, are explored. The deposit is considered to consist of two layers, fresh and aged; this simple picture is shown to be sufficient to interpret thermal and hydraulic aspects of deposit aging when HTDs are operated at constant heat flux (as reflecting laboratory experiments) but not in cases where the constant wall temperature approximation is more realistic.", "author" : [ { "dropping-particle" : "", "family" : "Bonis", "given" : "Ioannis", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Xie", "given" : "Weiguo", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Theodoropoulos", "given" : "Constantinos", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "AIChE JournalAIChE Journal", "id" : "ITEM-1", "issued" : { "date-parts" : [ [ "2012" ] ] }, "page" : "801-811", "title" : "A Linear Model Predictive Control Algorithm for Nonlinear Large-Scale Distributed Parameter Systems", "type" : "article-journal", "volume" : "58" }, "uris" : [ "" ] }, { "id" : "ITEM-2", "itemData" : { "DOI" : "10.1137/0730057", "ISBN" : "00361429", "ISSN" : "0036-1429", "PMID" : "17746741", "abstract" : "Fixed-point iterative procedures for solving nonlinear parameter dependent problems can converge for some interval of parameter values and diverge as the parameter changes. The Recursive Projection Method (RPM), which stabilizes such procedures by computing a projection onto the unstable subspace is presented. On this subspace a Newton or special Newton iteration is performed, and the fixed-point iteration is used on the complement. As continuation in the parameter proceeds, the projection is efficiently updated, possibly increasing or decreasing the dimension of the unstable subspace. The method is extremely effective when the dimension of the unstable subspace is small compared to the dimension of the system. Convergence proofs are given and pseudo-arclength continuation on the unstable subspace is introduced to allow continuation past folds. Examples are presented for an important application of the RPM in which a \u201cblack-box\u201d time integration scheme is stabilized, enabling it to compute unstable steady...", "author" : [ { "dropping-particle" : "", "family" : "Shroff", "given" : "Gautam M.", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Keller", "given" : "Herbert B.", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "SIAM Journal on Numerical Analysis", "id" : "ITEM-2", "issue" : "4", "issued" : { "date-parts" : [ [ "1993" ] ] }, "page" : "1099-1120", "title" : "Stabilization of Unstable Procedures: The Recursive Projection Method", "type" : "article-journal", "volume" : "30" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "^30,40", "manualFormatting" : "21,27", "plainTextFormattedCitation" : "30,40", "previouslyFormattedCitation" : "^30,40" }, "properties" : { }, "schema" : "" }21,ADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1016/j.ces.2011.09.033", "ISBN" : "0009-2509", "ISSN" : "00092509", "abstract" : "Most engineering systems can be accurately simulated using models consisting of Partial Differential Equations. Thus the challenging problem of PDE-constrained optimization arises naturally in engineering design. Issues surface due to the high number of variables involved and the use of specialized software for simulation which may not include an optimization option. In this work we present a methodology for the steady-state optimization of systems for which an input/output steady-state simulator is available. The proposed method is efficient for dissipative systems and is based on model reduction. This framework employs a two-step projection scheme, first onto the low-dimensional, adaptively computed, dominant subspace of the system and second onto the subspace of independent variables. Hence only low order Jacobian and Hessian matrices are used in this formulation, computed efficiently with directional perturbations. \u00a9 2011 Elsevier Ltd.", "author" : [ { "dropping-particle" : "", "family" : "Bonis", "given" : "Ioannis", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Theodoropoulos", "given" : "Constantinos", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Chemical Engineering Science", "id" : "ITEM-1", "issue" : "1", "issued" : { "date-parts" : [ [ "2012" ] ] }, "page" : "69-80", "publisher" : "Elsevier", "title" : "Model reduction-based optimization using large-scale steady-state simulators", "type" : "article-journal", "volume" : "69" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "²⁷", "plainTextFormattedCitation" : "27", "previouslyFormattedCitation" : "²⁷" }, "properties" : { }, "schema" : "" }27. The selection of the m slow modes is crucial for efficient model reduction and more details can be found in Bonis & TheodoropoulosADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1016/j.ces.2011.09.033", "ISBN" : "0009-2509", "ISSN" : "00092509", "abstract" : "Most engineering systems can be accurately simulated using models consisting of Partial Differential Equations. Thus the challenging problem of PDE-constrained optimization arises naturally in engineering design. Issues surface due to the high number of variables involved and the use of specialized software for simulation which may not include an optimization option. In this work we present a methodology for the steady-state optimization of systems for which an input/output steady-state simulator is available. The proposed method is efficient for dissipative systems and is based on model reduction. This framework employs a two-step projection scheme, first onto the low-dimensional, adaptively computed, dominant subspace of the system and second onto the subspace of independent variables. Hence only low order Jacobian and Hessian matrices are used in this formulation, computed efficiently with directional perturbations. \u00a9 2011 Elsevier Ltd.", "author" : [ { "dropping-particle" : "", "family" : "Bonis", "given" : "Ioannis", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Theodoropoulos", "given" : "Constantinos", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Chemical Engineering Science", "id" : "ITEM-1", "issue" : "1", "issued" : { "date-parts" : [ [ "2012" ] ] }, "page" : "69-80", "publisher" : "Elsevier", "title" : "Model reduction-based optimization using large-scale steady-state simulators", "type" : "article-journal", "volume" : "69" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "²⁷", "plainTextFormattedCitation" : "27", "previouslyFormattedCitation" : "²⁷" }, "properties" : { }, "schema" : "" }27. It should be noted that only the directional derivatives need to be computed as is the case of automatic differentiation (AD) approaches. As a result, AD can be combined with our model reduction approach to enhance even further the computational capabilities of our methodology. 3. MethodologyOur proposed methodologies combining equation-free model reduction with PRSQP to handle large-scale optimisation problems are presented below. Three different ways to handle (non-linear) inequality constraints are examined.3.1. Equation-Free Reduced PRSQP (EF-PRSQP)Here, equation-free model reduction is employed to project the full-scale system onto the low-dimensional space of the slow modes, taking advantage of the separation of scales, effectively reducing the dimensionality of the state variables given by the system equality constraints. The inequality constraints are handled as in PRSQP. Equation-free model reduction is successfully coupled with PRSQP using a 2-step projection schemeADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "Doi 10.1137/040604716", "ISBN" : "1540-3459", "ISSN" : "15403459", "abstract" : "A reduced model based optimization strategy is presented for the cases where input/ output codes are the process simulators of choice, and thus system Jacobians and even system equations are not explicitly available to the user. The former is the case when commercial software packages or legacy codes are used to simulate a large-scale system and the latter when microscopic or multiscale simulators are employed. When such black-box dynamic simulators are used, we perform optimization by combining the recursive projection method [ G. M. Shroff and H. B. Keller, SIAM J. Numer. Anal., 30 ( 1993), pp. 1099 - 1120] which identifies the ( typically) low-dimensional slow dynamics of the (dissipative) model with a second reduction to the low-dimensional subspace of the decision variables. This results in the solution of a low-order unconstrained optimization problem. Optimal conditions are then computed in an efficient way using only low-dimensional numerical approximations of gradients and Hessians. The tubular reactor is used as an illustrative example to demonstrate this model reduction-based optimization methodology.", "author" : [ { "dropping-particle" : "", "family" : "Luna-Ortiz", "given" : "E", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Theodoropoulos", "given" : "Constantinos", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Multiscale Modeling & Simulation", "id" : "ITEM-1", "issue" : "2", "issued" : { "date-parts" : [ [ "2005" ] ] }, "page" : "691-708", "title" : "An input/output model reduction-based optimization scheme for large-scale systems", "type" : "article-journal", "volume" : "4" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "³²", "plainTextFormattedCitation" : "32", "previouslyFormattedCitation" : "³²" }, "properties" : { }, "schema" : "" }32. In order to include the decision variables in the low-dimensional subspace, an extended orthonormal basis is defined:Zext=Z00Idof (10)Here Idof?R(N+dof)×dof is the identity matrix and dof is the number of decision (independent) variables. A coordinate basis of the subspace of the independent variables can be computed as:Zr=-H-1ZT?zGI (11)while a basis, Υ, for the complement subspace is given by: Υ=I0 (12)Hence, a projection basis, Z* , equivalent to the basis computed in rSQP5-8, can be calculated as: Z*=Zext Zr=-ZH-1ZT?zGI?R(N+dof)×dof (13)where only the inverse of the low-order matrix, H, is required. This projection is equivalent to the one of the reduced Hessian method, but not equal as the latter is the null space of the constraints and the former is essentially a double projection firstly on the low-order subspace of the dominant modes and subsequently on the null space of the constraints. Even though, both methods give the same basis size, rSQP requires the construction and inversion of a large matrix (Jacobian of constrains), whilst our model reduction-based method computes and inverts only a low-order matrix. The reduced Hessian, BR, is the restriction of the (unavailable) full system Hessian, B, on Z*ADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1016/j.ces.2011.09.033", "ISBN" : "0009-2509", "ISSN" : "00092509", "abstract" : "Most engineering systems can be accurately simulated using models consisting of Partial Differential Equations. Thus the challenging problem of PDE-constrained optimization arises naturally in engineering design. Issues surface due to the high number of variables involved and the use of specialized software for simulation which may not include an optimization option. In this work we present a methodology for the steady-state optimization of systems for which an input/output steady-state simulator is available. The proposed method is efficient for dissipative systems and is based on model reduction. This framework employs a two-step projection scheme, first onto the low-dimensional, adaptively computed, dominant subspace of the system and second onto the subspace of independent variables. Hence only low order Jacobian and Hessian matrices are used in this formulation, computed efficiently with directional perturbations. \u00a9 2011 Elsevier Ltd.", "author" : [ { "dropping-particle" : "", "family" : "Bonis", "given" : "Ioannis", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Theodoropoulos", "given" : "Constantinos", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Chemical Engineering Science", "id" : "ITEM-1", "issue" : "1", "issued" : { "date-parts" : [ [ "2012" ] ] }, "page" : "69-80", "publisher" : "Elsevier", "title" : "Model reduction-based optimization using large-scale steady-state simulators", "type" : "article-journal", "volume" : "69" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "²⁷", "plainTextFormattedCitation" : "27", "previouslyFormattedCitation" : "²⁷" }, "properties" : { }, "schema" : "" }27:BR=Z*TBZ* (14) Here B is the Hessian of the Lagrange function, Lx, defined as Lx=fx+GT(x)λ+hinT(x)λin. The reduced Hessian is efficiently computed taking advantage of the directional derivatives, employing the same central finite difference-based scheme as the one used for the reduced Jacobian (eq. 9). To accelerate this computation a BFGSADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1007/978-0-387-40065-5", "ISBN" : "1431-8598", "ISSN" : "10969101", "PMID" : "661941", "abstract" : "Summary: Numerical Optimization presents a comprehensive and up-to-date description of the most effective methods in continuous optimization. It responds to the growing interest in optimization in engineering, science, and business by focusing on the methods that are best suited to practical problems. For this new edition the book has been thoroughly updated throughout. There are new chapters on nonlinear interior methods and derivative-free methods for optimization, both of which are used widely in practice and the focus of much current research. Because of the emphasis on practical methods, as well as the extensive illustrations and exercises, the book is accessible to a wide audience. It can be used as a graduate text in engineering, operations research, mathematics, computer science, and business. It also serves as a handbook for researchers and practitioners in the field. The authors have strived to produce a text that is pleasant to read, informative, and rigorous - one that reveals both the beautiful nature of the discipline and its practical side.", "author" : [ { "dropping-particle" : "", "family" : "Nocedal", "given" : "Jorge", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Wright", "given" : "Stephen J", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Springer Series in Operations Research and Financial Engineering,", "id" : "ITEM-1", "issued" : { "date-parts" : [ [ "2006" ] ] }, "number-of-pages" : "XXII, 664 s. 85 illus.", "title" : "Numerical Optimization", "type" : "book" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "¹⁷", "plainTextFormattedCitation" : "17", "previouslyFormattedCitation" : "¹⁷" }, "properties" : { }, "schema" : "" }17 approach can be followed preserving positivity of the Hessian.It is important to mention that the dependent variables in the optimization problem satisfy the constraints at every iteration, so the procedure is a feasible-point algorithm. Thus, the reduced QP sub-problem is transformed as follows:minpZZ*T?fTpz+12pzTBRpzs.t. ?hxZ*pZ≤-h(x)xL-x≤Z*pz≤xU-x (15)Here pz?Rdof is the component of the search direction onto the subspace of the decision variables. The basis Z* and the reduced Hessian BR are given by eq. 13 and 14, respectively. The low-order projections, φ, of the Lagrange multipliers, λ ?R(N+dof)×dof, of the equality constraints onto P are computed as:HTφ=ZT(ΥΤ?f+λ inΤ?h) (16)where φ=ZTλ (17)It can be easily shown that the reduced optimization problem has the following KKT conditions:?fxT+HZTZT?zGxTφ+?hxTλin=0PGx=0 hiλini=0, for i=1…Nin (18)3.1.1 Proposed optimization schemeThe proposed method, which handles large-scale systems and nonlinear inequality constraints, is presented in Algorithm 1 ( REF _Ref454448814 \h \* MERGEFORMAT Table 1).Table 1. Optimization scheme including inequality constraints (EF-PRSQP), Algorithm 1Choose initial values for x Compute a feasible point using the black box simulator.Model reduction: Compute basis Z and reduced Jacobian H.Construct the basis Z* using eq. 13.If iteration number >1 and Z*pz>ε then compute the multipliers, φ, using values from the previous pute the reduced Hessian, BR, in order to solve the reduced QP problem minpZZk*T?fkTpz+12pzTBRpzs.t. ?hxZ*pZ≤-h(x)xL-x≤Z*pz≤xU-xUpdate the values of the variables: x=x+Z*pZ, xprevious=xIf Z*pz<ε update the basis Z*, and calculate the multipliers (eq. 16) based on the new reduced Jacobian, H.Hprevious=HCheck convergence (Z*pZ<ε); if the problem not converged then go to step 2Implementation of Algorithm 1 produces satisfactory results as is illustrated in the case study (see section 4), but has a major drawback: The number of inequality constraints may be large, because the corresponding physical system is distributed-parameter. Also, inequality constraints may hold for the whole spatial domain. Consequently, calculating the gradient of the inequality constraints with respect to all variables will be computationally expensive. 3.1.2 Equivalence of the computed optimaIn this section, we show that an optimum of the full problem is an optimum of the reduced problem (and vice-versa), if the reduced system is a good approximation of the full-scale one and the reduced states can accurately reproduce the full states. The KKT conditions of the full problem are the following:?f+?GTλ+?hTλin=0Gx=0 (19)hixλini=0 for some λin>0, λ>0. It can also be proven that our reduced optimization scheme has super-linear convergence properties. Theorem 1: For dissipative systems every optimum point satisfies eq. 19, iff satisfies eq. 18. Proof: Eq. 19 can equivalently be written as:?fT+P+Q?GPext+QextTλ+?hTλin=0?fT+P?GPextTλ+P?GQext+Q?GPext+Q?GQextTλ+?hTλin=0 (20)Taking into account eq.17?fT+Pext?GTZφ+P?GQext+Q?GPext+Q?GQextTλ+?hTλin=0 (21)We can set P?GQext+Q?GPext+Q?GQextTλ=E (22)E being the error associated with the model reduction operation. Therefore, eq. (21) can be written as: ?fT+Pext?GTZφ+E+?hTλin=0 (23)This can be easily shown to be equivalent to: ?fT+HZTZT?zGxTφ+E+?hTλin=0 (24)As long as the basis, Z?, approximates the maximal invariant subspace of ?uG and the dominant modes are captured by the model reduction with x≈Z?vADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1137/0730057", "ISBN" : "00361429", "ISSN" : "0036-1429", "PMID" : "17746741", "abstract" : "Fixed-point iterative procedures for solving nonlinear parameter dependent problems can converge for some interval of parameter values and diverge as the parameter changes. The Recursive Projection Method (RPM), which stabilizes such procedures by computing a projection onto the unstable subspace is presented. On this subspace a Newton or special Newton iteration is performed, and the fixed-point iteration is used on the complement. As continuation in the parameter proceeds, the projection is efficiently updated, possibly increasing or decreasing the dimension of the unstable subspace. The method is extremely effective when the dimension of the unstable subspace is small compared to the dimension of the system. Convergence proofs are given and pseudo-arclength continuation on the unstable subspace is introduced to allow continuation past folds. Examples are presented for an important application of the RPM in which a \u201cblack-box\u201d time integration scheme is stabilized, enabling it to compute unstable steady...", "author" : [ { "dropping-particle" : "", "family" : "Shroff", "given" : "Gautam M.", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Keller", "given" : "Herbert B.", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "SIAM Journal on Numerical Analysis", "id" : "ITEM-1", "issue" : "4", "issued" : { "date-parts" : [ [ "1993" ] ] }, "page" : "1099-1120", "title" : "Stabilization of Unstable Procedures: The Recursive Projection Method", "type" : "article-journal", "volume" : "30" }, "uris" : [ "" ] }, { "id" : "ITEM-2", "itemData" : { "author" : [ { "dropping-particle" : "", "family" : "Theodoropoulos", "given" : "C", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Lecture Notes in Computational Science and Engineering Vol 75", "id" : "ITEM-2", "issued" : { "date-parts" : [ [ "2011" ] ] }, "page" : "37-61", "title" : "Optimisation and linear control of large scale nonlinear systems: A review and a suite of model reduction-based techniques", "type" : "chapter" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "^30,31", "plainTextFormattedCitation" : "30,31", "previouslyFormattedCitation" : "^30,31" }, "properties" : { }, "schema" : "" }30,31, E is small and bounded. In addition, Z is udated at every iteration of the algorithm, to ensure that the full-scale model is adequately captured by the dominant modes. Hence E→0, throughout, and the following holds:?fT+HZTZT?zGxTφ+E+?hTλin=0 (25)In addition, PG(x)=0 holds, because the (feasible point) algorithm uses a black-box simulator, which solves the equality constraints. Also hxλin=0 holds, as the inequality constraints are part of the solution of the QP problem at every iteration. Hence the equivalence of eq. (19) with eq. (18) is proven. This of course does not guarantee that the reduced problem does not exhibit additional stationary points which satisfy the KKT conditions. The inverse can be shown accordingly. The KKT conditions corresponding to Algorithm 1 (eq. 18) can be written as: ?fT+Pext?GTZφ+?hTλin=0?fT+Pext?GTPλ+?hTλin=0 (26)The term Pext?GTPλ from eq. (26) is equal to ?Gλ-E. As above we can assume that E→0 when the full-scale model is adequately captured by the dominant modes. Additionally, since the algorithm is feasible-path, G(x)= 0, and PGx=0. Also, hxλin=0 as explained above. Then (eq. 26) becomes equal to (eq. 19) ?3.2. Equation-Free Reduced PRSQP with Aggregated Inequalities (EF-PRSQP-KS)Equation-Free Model Reduced PRSQP handles the inequality constraints effectively. Nevertheless, the derivatives of all inequality constraints are needed, which means that the computational efficiency may be jeopardised, since conditions, such as safety specifications or economic restrictions, may be applied to the whole spatial domain. An effective way to tackle this problem is to use an aggregation function, such as the KS function in eq. 3, which can be combined with the Equation-Free PRSQP in order to produce an efficient optimization algorithm suitable for a large number of both equality and inequality constraints. The main advantage of this KS aggregation function is the ability to substitute all the inequality constraints with only one. It can easily be shown that if all the inequality constraints are negative then the KS function is negative as well, and also if there is an active set of inequality constraints then KS approximates zero as ρ→∞ADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1016/S0098-1354(00)00591-3", "ISSN" : "00981354", "abstract" : "Process flexibility and design under uncertainty have been researched extensively in the literature. Problem formulations for flexibility include nested optimization problems and these can often be refined by substituting the optimality conditions for these nested problems. However, these reformulations are highly constrained and can be expensive to solve. In this paper we extend algorithms to solve these reformulated NLP problem under uncertainty by introducing two contributions to this approach. These are the use of a Constraint Aggregation function (KS function) and Smoothing Functions. We begin with basic properties of KS function. Next, we review a class of parametric smooth functions, used to replace the complementarity conditions of the KKT conditions with a well-behaved, smoothed nonlinear equality constraint. In this paper we apply the previous strategies to two specific problems: i) the?worst case algorithm?, that assesses design under uncertainty and, ii) the flexibility index and feasibility test formulations. In the first case, two new algorithms are derived, one of them being single level optimization problem. Next using similar ideas, both flexibility index and feasibility test are reformulated leading to a single non linear programming problem instead of a mixed integer non linear programming one. The new formulations are demonstrated on five different example problems where a CPU time reduction of more than 70 and 80% is demonstrated.", "author" : [ { "dropping-particle" : "", "family" : "Raspanti", "given" : "C G", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Bandoni", "given" : "J a", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Biegler", "given" : "L T", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Computers & Chemical Engineering", "id" : "ITEM-1", "issue" : "9-10", "issued" : { "date-parts" : [ [ "2000" ] ] }, "page" : "2193-2209", "title" : "New strategies for flexibility analysis and design under uncertainty", "type" : "article-journal", "volume" : "24" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "²¹", "plainTextFormattedCitation" : "21", "previouslyFormattedCitation" : "²¹" }, "properties" : { }, "schema" : "" }21. It is then easy to replace the formulation of the nonlinear optimization problem of eq. 6 with the formulation of eq. 27minfxs.t. Gx=0 (27)KSx,ρ≤0Yet, this formulation may produce sub-optimal results, and the solution gets closer to the optimum for high values of ρ. This behaviour can be explained since the KS function creates a smaller feasible region for the optimizer than the original one. Nevertheless, the parameter ρ cannot be too large from the beginning of the algorithm, because numerical difficulties may arise. As a result, an adaptive procedure should be implemented. Poon and MartinsADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "author" : [ { "dropping-particle" : "", "family" : "Poon", "given" : "Nmk", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Martins", "given" : "J", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "2005 Canadian Aeronautics and Space Institute Annual General Meeting", "id" : "ITEM-1", "issued" : { "date-parts" : [ [ "2005" ] ] }, "page" : "1-12", "title" : "Adaptive Constraint Aggregation for Structural Optimization Using Adjoint Sensitivities", "type" : "article-journal" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "²⁰", "plainTextFormattedCitation" : "20", "previouslyFormattedCitation" : "²⁰" }, "properties" : { }, "schema" : "" }20 introduce such an adaptive approach in order to avoid sub-optimal results. In this approach the aggregation parameter, ρ, changes according to the sensitivity of the KS function. The aggregation parameter is increased so that the derivative of KS, KS', is less than (or equal to) a small number. This number can be defined as a desired value KSd'. Hence, assuming that KS' has a linear dependence on the aggregation parameterADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "author" : [ { "dropping-particle" : "", "family" : "Poon", "given" : "Nmk", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Martins", "given" : "J", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "2005 Canadian Aeronautics and Space Institute Annual General Meeting", "id" : "ITEM-1", "issued" : { "date-parts" : [ [ "2005" ] ] }, "page" : "1-12", "title" : "Adaptive Constraint Aggregation for Structural Optimization Using Adjoint Sensitivities", "type" : "article-journal" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "²⁰", "plainTextFormattedCitation" : "20", "previouslyFormattedCitation" : "²⁰" }, "properties" : { }, "schema" : "" }20, a relationship between the current value, ρc, of the aggregation parameter and the desired one ρd, can be found:logKS1'-logKSc'ρ1-ρc=logKSd'-logKSc'ρd-ρc (28)ρ1 being the value of the parameter at a small step ahead. Solving eq. 28 with respect to ρd the following equation is derived: ρd=exp?(logKSd'KSc'logKS1'KSc'-1logρ1ρc+logρc) (29)In algorithm 2 ( REF _Ref453858228 \h \* MERGEFORMAT Table 2) an adaptive procedure is presentedADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "author" : [ { "dropping-particle" : "", "family" : "Poon", "given" : "Nmk", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Martins", "given" : "J", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "2005 Canadian Aeronautics and Space Institute Annual General Meeting", "id" : "ITEM-1", "issued" : { "date-parts" : [ [ "2005" ] ] }, "page" : "1-12", "title" : "Adaptive Constraint Aggregation for Structural Optimization Using Adjoint Sensitivities", "type" : "article-journal" }, "uris" : [ "" ] }, { "id" : "ITEM-2", "itemData" : { "abstract" : "This paper explores different methods of constraint aggregation for numerical optimization. The main motive is the aggregation of stress constraints in structural weight minimization problems in order to reduce the cost of adjoint sensitivity calculations and hence the overall cost of the optimization. We analyze existing approaches such as considering all constraints individually, taking the maximum of the constraints and using the Kreisselmeier\u2013Steinhauser function. A new adaptive approach based on the Kreisselmeier\u2013Steinhauser function is proposed and is shown to significantly increase the accuracy of the results when a large number of constraints is active at the optimum.", "author" : [ { "dropping-particle" : "", "family" : "Martins", "given" : "Joaquim R R a", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Poon", "given" : "Nicholas M K", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Proceedings of 6th World Congress on Structural and Multidisciplinary Optimization", "id" : "ITEM-2", "issue" : "June", "issued" : { "date-parts" : [ [ "2005" ] ] }, "page" : "1-10", "title" : "On Structural Optimization Using Constraint Aggregation", "type" : "article-journal" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "^20,34", "plainTextFormattedCitation" : "20,34", "previouslyFormattedCitation" : "^20,34" }, "properties" : { }, "schema" : "" }20,34, taking into account the constraint functions, the current aggregation parameter ρc and the desired sensitivity. Table 2. Adaptation procedure of aggregation parameter Algorithm 2Compute the derivative of KS (KS') at the current pointIf the current value is smaller than the desired, return the current valueOtherwise compute ρd according to eq. 29Compute KS using ρdThe aggregation function allows us to handle all inequality constraints with a single constraint. Consequently, the computational time will decrease dramatically, as the case study will show. 3.2.1 Proposed optimization scheme. In Algorithm 3 ( REF _Ref453858019 \h \* MERGEFORMAT Table 3) a modification of Algorithm 1 ( REF _Ref454448814 \h \* MERGEFORMAT Table 1) is presented, including the adaptation procedure for the aggregation function. This algorithm is implemented in section 4, where all approaches are applied to a chemical engineering example and are evaluated and compared. Table 3. EF-PRSQP-KS Algorithm 3Steps 1. -5. are the same with Algorithm 1Compute the reduced HessianADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1016/j.ces.2011.09.033", "ISBN" : "0009-2509", "ISSN" : "00092509", "abstract" : "Most engineering systems can be accurately simulated using models consisting of Partial Differential Equations. Thus the challenging problem of PDE-constrained optimization arises naturally in engineering design. Issues surface due to the high number of variables involved and the use of specialized software for simulation which may not include an optimization option. In this work we present a methodology for the steady-state optimization of systems for which an input/output steady-state simulator is available. The proposed method is efficient for dissipative systems and is based on model reduction. This framework employs a two-step projection scheme, first onto the low-dimensional, adaptively computed, dominant subspace of the system and second onto the subspace of independent variables. Hence only low order Jacobian and Hessian matrices are used in this formulation, computed efficiently with directional perturbations. \u00a9 2011 Elsevier Ltd.", "author" : [ { "dropping-particle" : "", "family" : "Bonis", "given" : "Ioannis", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Theodoropoulos", "given" : "Constantinos", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Chemical Engineering Science", "id" : "ITEM-1", "issue" : "1", "issued" : { "date-parts" : [ [ "2012" ] ] }, "page" : "69-80", "publisher" : "Elsevier", "title" : "Model reduction-based optimization using large-scale steady-state simulators", "type" : "article-journal", "volume" : "69" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "²⁷", "plainTextFormattedCitation" : "27", "previouslyFormattedCitation" : "²⁷" }, "properties" : { }, "schema" : "" }27 in order to solve the reduced QP problemCompute the KS function and its derivativeApply the Algorithm 2 ( REF _Ref453858228 \h \* MERGEFORMAT Table 2)Solve the reduced QP problem and calculate the Lagrange multipliers for the inequality constraints minpZZk*T?fkTpz+12pzTBRpzs.t. KS'x,ρZ*pZ≤-KSx,ρxL-x≤Z*pz≤xU-xSteps 7. -10. are the same with Algorithm 13.2.2 Equivalence of the computed optima In this section it is shown that every optimization point of the full NLP problem is also an optimum of the reduced problem when the aggregation function is applied. If the reduced space is a good approximation of the real one, the reduced states can accurately reconstruct the full states and also the adaptive procedure of the aggregation function produces a large enough ρ when required. Then Theorem 1 can be applied to prove the equivalence of the computed optima. An additional fair assumption has been posed here: The aggregation parameter ρ, is large enough and can be produced by the adaptation procedure in Algorithm 2. 3.3. Equation-Free Reduced PRSQP with Aggregated Inequalities and Slack Variable (EF-PRSQP-KS-S)Slack variables have been utilized to handle inequality constraints; however the straightforward approach of introducing slack variables and treating inequality constraints as equalities is not a reasonable approach when the methodology implemented, includes model reduction or generally partitioning of the solution space. If the model reduction includes active inequality constraints then 2 undesirable effects may arise: The dimension of the basis would vary during run-time, depending on the number of active inequality constraints. Then it will be difficult for the algorithm to give a good initial guess for the basis for the dominant subspace. Hence the numerical efficiency will be jeopardized.If the eigenvalues of the active inequality constraints are aggregated with the eigen-spectrum of the equality constraints then this will ruin the separation of scales due to the addition of non-dissipative modes coming from the active inequality constraints.These disadvantages are overcome using an aggregation function, as only one equality will be added. The KS function, presented in section 2, can aggregate effectively all the inequality constraints. If there is only one inequality in the problem then the eigen-spectrum will not change significantly when slack variables are added. This is proven in Lemma 1. Lemma 1:If only one inequality is added to the problem then only one eigenvalue equal to one will be added in the eigenspectrum of the original problem. Proof:Gx=0h1x≤0 slack variables hnew=h1x+s=0 (30)where s is the slack variable that is zero when the inequality constraints are active and positive otherwise. The Jacobian of the augmented system is: Jaug=?xGT?sGT?xhnew?sh (31)The derivative of G with respect to the slack variable is always zero and the derivative of hnew is always one. Furthermore, the derivative of hnew with respect to the states is equal to the derivative of the inequality constraints. As a result the augmented Jacobian can be written asJaug=?xGT??xh1 (32)Every eigenvalue should be the solution of the following problem.detJaug-λeigIN+1=0 (33)det?xGT??xh1-λeigIN+1=0det?xGT-λeigIN??xh1-λeig=0-1N+2?h?x1det?G1?x1-λeig?0????GN ?x1?0 +…+1-λeigdet(?xGT-λeigIN) =0 (34)In eq. 34 all the determinants are zero because all of them have (at least) one zero column, except from the last one. So, eq. 34 can be rewritten as1-λeigdet(?xGT-λeigIN)=0 (35)The solution of eq. 35 produces the original eigenvalues of the system and/or one eigenvalue equal to one. ?The lemma shows that a slack variable can be used without a significant disturbance in the eigen-spectrum of the system. Specifically, to ensure that no critical eigenvalue will be overlooked, the dominant space enlarged by one (m+1). 3.3.1 Proposed optimization schemeIn this optimization algorithm, model reduction proceeds as in Algorithm 1, the adaptive KS function must be calculated in every step as in Algorithm 3 (Table 3). A slack variable is introduced and the solver solves an additional equation. The algorithm is shown in Table 4.Table 4. EF-slack variables-KS-S Algorithm 4Step 1. – 2. are the same with Algorithm 3Use Algorithm 2 to find the adaptive KS Find the slack variableStep 3. – 5. are the same with Algorithm 1Compute the reduced HessianADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1016/j.ces.2011.09.033", "ISBN" : "0009-2509", "ISSN" : "00092509", "abstract" : "Most engineering systems can be accurately simulated using models consisting of Partial Differential Equations. Thus the challenging problem of PDE-constrained optimization arises naturally in engineering design. Issues surface due to the high number of variables involved and the use of specialized software for simulation which may not include an optimization option. In this work we present a methodology for the steady-state optimization of systems for which an input/output steady-state simulator is available. The proposed method is efficient for dissipative systems and is based on model reduction. This framework employs a two-step projection scheme, first onto the low-dimensional, adaptively computed, dominant subspace of the system and second onto the subspace of independent variables. Hence only low order Jacobian and Hessian matrices are used in this formulation, computed efficiently with directional perturbations. \u00a9 2011 Elsevier Ltd.", "author" : [ { "dropping-particle" : "", "family" : "Bonis", "given" : "Ioannis", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Theodoropoulos", "given" : "Constantinos", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Chemical Engineering Science", "id" : "ITEM-1", "issue" : "1", "issued" : { "date-parts" : [ [ "2012" ] ] }, "page" : "69-80", "publisher" : "Elsevier", "title" : "Model reduction-based optimization using large-scale steady-state simulators", "type" : "article-journal", "volume" : "69" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "²⁷", "plainTextFormattedCitation" : "27", "previouslyFormattedCitation" : "²⁷" }, "properties" : { }, "schema" : "" }27 in order to solve the reduced QP problemminpZ(Z*T?f)Tpz+12pzTBRpzxL-x≤Z*pz≤xU-xSteps 7. – 10. are the same with Algorithm 13.3.2 Equivalence of computed optimaThe approach in this section aggregates all the inequality constraints into one KS function, and then a slack variable is used to transform the resulting inequality constraint into an equality constraint. According to Lemma 1, the eigenspectrum will not be disturbed when the KS function is used in conjunction with a slack variable. Therefore, Theorem 1 can be used to show that the computed optimum of the reduced problem is also an optimum problem of the full NLP.4. ResultsTo illustrate the behaviour of the proposed optimization algorithms, a case study of a tubular reactor is implemented. In the reactor an exothermic, first order, irreversible reaction takes place (A→B). The reactor has three heat exchangers on its jacket, the temperature in each heat exchanger is considered to be constant and is used as a degree of freedom for the optimization problem. The model of the reactor consists of 2 PDEsADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "abstract" : "Abatracf-Methods for studying the bifurcation behavior of tubular reactors have been developed. This involves the application of static and Hopf bifurcation theory for PDE's and the very precise determination of steady state profiles. Practical computational methods for carrying out this analysis are discussed in some detail. For the special case of a first order, irreversible reaction in a tubular reactor with axial dispersion, the bifurcation behavior is classified and summarized in parameter space plots. In particular the influence of the Lewis and Peclet numbers is investigated. It is shown that oscillations due to interaction of dispersion and reaction effects should not exist in fixed bed reactors and moreover, should only occur in very short \" empty \" tubular reactors. The parameter study not only brings together previously published examples of multiple and periodic solutions but also reveals a hitherto undiscovered wealth of bifurcation slructures. Sixteen of these structures, which come about by combinations of as many as four bifurcations to multiple steady states and four bifurcations to periodic solutions, are ihusttated with numerical examples. Although the analysis is based on the pseudohomogeneous axial dispersion model, it can readily be applied to other reaction diffusion equations such as the general two phase models for fixed bed reactors.", "author" : [ { "dropping-particle" : "", "family" : "Jensen", "given" : "Klavs F", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Harmon Ray", "given" : "W", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Chemical Engheetinrhg Science", "id" : "ITEM-1", "issue" : "2", "issued" : { "date-parts" : [ [ "1982" ] ] }, "page" : "199-222", "title" : "The bifurcation behaviour of tubular reactors", "type" : "article-journal", "volume" : "37" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "⁴²", "plainTextFormattedCitation" : "42", "previouslyFormattedCitation" : "⁴²" }, "properties" : { }, "schema" : "" }42 (Note that in this case the system of equations 36-37 is actually a set of ODEs; however we keep the term PDE as the methodology does not change for problems with more partial derivatives, e.g. 2- or 3-dimensional systems):1Pe1?2x1?y2-?x1?y+Da1-x1expx21+x2γ=0(36) 1LePe2?2x2?y2-1Le?x2?y+CLeDa1-x1expx21+x2γ+βLe(x2w-x2)=0 (37)where is the dimensionless concertation of the product, the dimensionless temperature inside the reactor, the Damkohler number, the Lewis number, Pe1 and Pe2 the Peclet numbers for mass and heat transfer respectively, the dimensionless heat transfer, C the dimensionless adiabatic temperature rise, the dimensionless activation energy, ∈[0,L] the dimensionless longitude coordinate, L the length of the reactor and the dimensionless adiabatic wall temperature. , whose expression is given as a function of the longitudinal coordinate: (38)where is the Heaviside function, , , and the dimensionless temperature at each of the 3 cooling zones. The boundary conditions are: (39) (40)The parameters of the physical model are and L=1. The set of equations (36-37) are discretised with the central Finite Differences method on a mesh of 250 nodes. This discretisation results in n = 500 dependent variables. The size of the subspace, m, is chosen to be 10 so it can be large enough to capture the dominant dynamics throughout the parameter space.The optimization problem aims to maximize the concertation at the outlet of the reactor by changing the 3 wall temperatures. The basic problem consists of only equality constraints the steady state mass and energy balancesADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1016/j.ces.2011.09.033", "ISBN" : "0009-2509", "ISSN" : "00092509", "abstract" : "Most engineering systems can be accurately simulated using models consisting of Partial Differential Equations. Thus the challenging problem of PDE-constrained optimization arises naturally in engineering design. Issues surface due to the high number of variables involved and the use of specialized software for simulation which may not include an optimization option. In this work we present a methodology for the steady-state optimization of systems for which an input/output steady-state simulator is available. The proposed method is efficient for dissipative systems and is based on model reduction. This framework employs a two-step projection scheme, first onto the low-dimensional, adaptively computed, dominant subspace of the system and second onto the subspace of independent variables. Hence only low order Jacobian and Hessian matrices are used in this formulation, computed efficiently with directional perturbations. \u00a9 2011 Elsevier Ltd.", "author" : [ { "dropping-particle" : "", "family" : "Bonis", "given" : "Ioannis", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Theodoropoulos", "given" : "Constantinos", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Chemical Engineering Science", "id" : "ITEM-1", "issue" : "1", "issued" : { "date-parts" : [ [ "2012" ] ] }, "page" : "69-80", "publisher" : "Elsevier", "title" : "Model reduction-based optimization using large-scale steady-state simulators", "type" : "article-journal", "volume" : "69" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "²⁷", "plainTextFormattedCitation" : "27", "previouslyFormattedCitation" : "²⁷" }, "properties" : { }, "schema" : "" }27. In reactors that exothermic reactions take place, a thermal runawayADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1016/j.jlp.2016.04.004", "ISSN" : "09504230", "abstract" : "Thermal safety and risk of accidents are still challenging topics in the case of batch reactors carrying exothermic reactions. In the present paper, the authors develop an integrated framework focusing on defining the governing parameters for the thermal runaway and evaluating the subsequent risk of accident. A relevant set of criteria are identified in order to find the prior conditions for a thermal runaway: failure of the cooling system, critical temperature threshold, successive derivatives of the temperature (first and second namely) vs. time and no detection in due time (reaction time) of the runaway initiation. For illustrative purposes, the synthesis of peracetic acid (PAA) with hydrogen peroxide (HP) and acetic acid (AA) is considered as case study. The critical and threshold values for the runaway accident are identified for selected sets of input data. Under the conditional probability of prior cooling system failure, Monte Carlo simulations are performed in order to estimate the risk of thermal runaway accident in batch reactors. It becomes then possible to predict the ratio of reactors, within an industrial plant, potentially subject to thermal runaway accident.", "author" : [ { "dropping-particle" : "", "family" : "Ni", "given" : "Lei", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Mebarki", "given" : "Ahmed", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Jiang", "given" : "Juncheng", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Zhang", "given" : "Mingguang", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Pensee", "given" : "Vincent", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Dou", "given" : "Zhan", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Journal of Loss Prevention in the Process Industries", "id" : "ITEM-1", "issued" : { "date-parts" : [ [ "2016", "9" ] ] }, "page" : "75-82", "title" : "Thermal risk in batch reactors: Theoretical framework for runaway and accident", "type" : "article-journal", "volume" : "43" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "⁴³", "plainTextFormattedCitation" : "43", "previouslyFormattedCitation" : "⁴³" }, "properties" : { }, "schema" : "" }43 may produce uncontrollable situations. To control this phenomenon, inequality constraints should be applied to meet some safety specifications. In this case study the dimensionless reaction rate is selected to have an upper-bound in order to avoid thermal explosions. The optimization problem is then set up as:maxx2wx1(y=1) (41)s.t. Gx=0hx≤3.50≤x1≤10≤x2≤80≤x2wi≤4i=1,…,3where is the dimensionless reaction rate across the reactor, given by eq. 42, are the PDEs of the physical model (eq. 36,37) in discretized form and x=x1T,x2T,x2wiTT . (42)The inequality constraints, h(x), are applied to the whole length of the reactor to ensure the limit is not surpassed at any point. Thus, not only the problem consists of a large number of equality constraints but also of a large number of inequality constraints. All three approaches produce the same optimal solution and in order to illustrate and compare the 3 optimization algorithms, the methodologies are evaluated in terms of the number of iterations for the same initial guess and CPU-time for different initial guesses. The initial guess for the Equation Free-PRSQP-KS and the Equation Free -PRSQP should satisfy the inequality constraints. Firstly, a base case scenario is presented in order to observe the optimization path and the general convergence behavior, where the initial guess for all wall temperatures is 0.2. Convergence results are given in terms of Z*pz, which is the norm of the solution update for each iteration. As mentioned before, the optimum point computed for all three algorithms is the same. The results presented in Figures 2-3 depict the solution of the problem with inequality constraints alongside with the solution of the problem taking in account only equality constraints (Initial problem). As it can be seen in Figure 3, the reaction rate and the temperature of the inequality-constrained problem are higher than those of the initial problem beyond y=0.2 to allow the reactant concentration to reach the optimum point. Figure 2. Dimensionless concentration at the optimum point with and without inequality constraints. Figure 3. (a) Dimensionless reaction rate and (b) dimensionless temperatures at the optimum point with and without inequality constraints.Despite the fact that all 3 methods produce identical optimum point results, the convergence path and computational time for each of the methods varies. Convergence paths for the three approaches are depicted in Figure 4a showing that the norm of the solution follows almost the same path for each method for the same initial guess ( dimensionless temperature all 3 cooling zones being 0.2) but Equation Free-PRSQP-KS requires the minimum amount of iterations, while Equation Free-PRSQP-KS-S requires the most. It is important to have a knowledge on how an algorithm convergences for different initial guesses, because in most cases there is no a priori knowledge of good guesses. In order to examine this behavior, 6 experiments have been performed for every algorithm: Initially, all wall temperatures (degrees of freedom) start with the value 0.00, then the code runs and the CPU-time is recorded. After that, wall temperatures are given the previous initial guess value plus 0.075, and the algorithm starts again. These experiments are conducted for all the proposed algorithms. The corresponding results are depicted in Figure 4b.Figure 4. (a) Convergence behaviour for each method starting from the same initial guess. (b) CPU- times for each method for a range of initial guesses for the degrees of freedom. As it can be seen, EF-PRSQP-KS is the fastest with the minimum number of iterations, whilst the slowest is the EF-PRSQP as expected. Figure 4 shows that EF-PRSQP-KS is faster in comparison to the others, for almost all the different initial guesses tested. In conclusion, all three algorithms seem to have the same trend in terms of iterations and computational time for all the experiments tested, however the fastest algorithm regardless of the number of iterations is the EF-PRSQP-KS, EF-PRSQP-KS-S comes next, and the slowest is EF-PRSQP. Thus, the aggregation of the inequality constraints into one seems to have a significant impact into the computational time as the CPU time of EF-PRSQP-KS is always around 80% faster than that of EF-PRSQP. To investigate how the fastest method, EF-PRSQP-KS, scales with problem size, we have tested the solution of the same system (eq. 36-42) for n=1000 and 1500, respectively. In addition, we have compared its performance (in CPU s) against that of PRSQP and of a “standard” NAG solver. As we can see in Table 1, EF-PRSQP-KS is approximately 8-10 times faster than PRSQP for all problem sizes and 17- 58 times faster than standard NAG-based SQP which solves the full model. All 3 methods converge to the same solution. Table 1: Comparison of performance of EF_PRSQP-KS against PRSQP and NAG SQP for different problem sizes. Problem size, nEF-PRSQP-KS, m=10(CPU, s)PRSQP(CPU, s)NAG SQP(CPU, s)5008.864150100039.15091293150075.87964454In addition, to investigate the effect of the size of m, we have tested the performance of EF_PRSQP-KS for m ranging from 10 to 50 for system size n = 500 and 1000. As it can be seen in Table 2, the method performs equally well for the whole range of m, which attests to its robustness. It is worthwhile to note that the method is even faster for m = 20-50 than for m =10 as the system requires less iterations to converge as seen in Fig. 5, where the convergence behavior is plotted for m = 10-30. Table 2: Comparison of performance of EF_PRSQP-KS for different subspace, m, sizes. Problem size(number of nodes, n)m=10(CPU, s)m=20(CPU, s)m=30(CPU, s)m=40(CPU, s)m=50(CPU, s)5008.85.55.65.86.1100039.12223.325.325.6Figure 5. Convergence behaviour of EF_PRSQP-KS from the same initial guess for different subspace, m, sizes. The Arnoldi iterations as well as the solution of each sub-QP and SQP-based solutions were computed using Nag library Mark 25 in Nag Builder 6.1.The times reported correspond to single threaded executions of FORTRAN source code on an Intel core i7-6700 processor (3.40 GHz) and 16 GB of RAM, running 64-bit windows 7. 5. Conclusions A model reduction-based deterministic optimisation method for large-scale nonlinear dissipative PDE‐constrained problems, which involve a large number of nonlinear inequality constraints has been presented. The work builds on previous research from the group presented earlierADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1016/j.ces.2011.09.033", "ISBN" : "0009-2509", "ISSN" : "00092509", "abstract" : "Most engineering systems can be accurately simulated using models consisting of Partial Differential Equations. Thus the challenging problem of PDE-constrained optimization arises naturally in engineering design. Issues surface due to the high number of variables involved and the use of specialized software for simulation which may not include an optimization option. In this work we present a methodology for the steady-state optimization of systems for which an input/output steady-state simulator is available. The proposed method is efficient for dissipative systems and is based on model reduction. This framework employs a two-step projection scheme, first onto the low-dimensional, adaptively computed, dominant subspace of the system and second onto the subspace of independent variables. Hence only low order Jacobian and Hessian matrices are used in this formulation, computed efficiently with directional perturbations. \u00a9 2011 Elsevier Ltd.", "author" : [ { "dropping-particle" : "", "family" : "Bonis", "given" : "Ioannis", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Theodoropoulos", "given" : "Constantinos", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Chemical Engineering Science", "id" : "ITEM-1", "issue" : "1", "issued" : { "date-parts" : [ [ "2012" ] ] }, "page" : "69-80", "publisher" : "Elsevier", "title" : "Model reduction-based optimization using large-scale steady-state simulators", "type" : "article-journal", "volume" : "69" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "²⁷", "plainTextFormattedCitation" : "27", "previouslyFormattedCitation" : "²⁷" }, "properties" : { }, "schema" : "" }27. The proposed algorithms are based on a 2 step-projection scheme. The first projection is onto the dominant space of the physical system, which is defined by the user in the beginning of the algorithm, augmented in order to include the independent variables. Then the second projection is onto the null space of the equality constraints. In this work all the methodologies make the assumption that the system is dissipative and thus there exists a separation of scales. As a result, there exists a low-order basis, which can be efficiently computed in order to approximate the full large-scale system. Three approaches have been followed to handle (the large number of) inequality constraints: the Equation-free-reduced-PRSQP method that reduces the number of equality constraints and then includes the inequality constraints in the quadratic sub-problem, This approach may exhibit numerical difficulties as the full derivatives of the inequality constraints with respect to the states have to be computed. The second approach takes advantage of the KS aggregation functions in order to handle only one inequality and combines it with the former method. The last approach uses slack variables to convert the inequality constraints into equalities. This is convenient when an aggregated function is used because the eigenvalue of the one additional equality is known and is equal to one. The behaviour of the three approaches is illustrated by applying them to the optimisation of a tubular reactor also testing the behaviour of the three algorithms for different initial guesses. In conclusion, the EF-PRSQP-KS seems to have the best performance, as it requires the least computational time and iterations compared to the other two. This methodology, was applied to a static system assuming that there is an available steady state simulator, however a dynamic model where Gx=xk+1-gxk=0 can be available instead. In this case the method presented in Theodoropoulos and Luna-OrtizADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "author" : [ { "dropping-particle" : "", "family" : "Theodoropoulos", "given" : "C", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Luna-ortiz", "given" : "Eduardo", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena", "id" : "ITEM-1", "issued" : { "date-parts" : [ [ "2006" ] ] }, "page" : "535-560", "title" : "A Reduced Input / Output Dynamic Optimisation Method", "type" : "chapter" }, "uris" : [ "" ] } ], "mendeley" : { "formattedCitation" : "³³", "plainTextFormattedCitation" : "33", "previouslyFormattedCitation" : "³³" }, "properties" : { }, "schema" : "" }33 can be extended to include inequalities.Acknowledgements The EU program CAFE (KBBE-2008-212754) and the University of Manchester Presidential Doctoral Scholarship Award to Panagiotis Petsagkourakis are gratefully acknowledged. 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Loss Prev. Process Ind. 2016, 43, 75.Table of Contents (TOC) Graphical Abstract ................
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