Linear Programming Lecture Notes
Linear Programming: Penn State Math 484 Lecture Notes
Version 1.8.3
Christopher Griffin 2009-2014
Licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License
With Contributions By: Bob Pakzad-Hurson Greg Ference Veselka Kafedzhieva Michael Cline Akinwale Akinbiyi Ethan Wright Richard Benjamin Douglas Mercer
Contents
List of Figures
v
Preface
ix
Chapter 1. Introduction to Optimization
1
1. A General Maximization Formulation
2
2. Some Geometry for Optimization
4
3. Gradients, Constraints and Optimization
10
Chapter 2. Simple Linear Programming Problems
13
1. Modeling Assumptions in Linear Programming
14
2. Graphically Solving Linear Programs Problems with Two Variables (Bounded
Case)
16
3. Formalizing The Graphical Method
17
4. Problems with Alternative Optimal Solutions
18
5. Problems with No Solution
20
6. Problems with Unbounded Feasible Regions
22
Chapter 3. Matrices, Linear Algebra and Linear Programming
27
1. Matrices
27
2. Special Matrices and Vectors
29
3. Matrices and Linear Programming Expression
30
4. Gauss-Jordan Elimination and Solution to Linear Equations
33
5. Matrix Inverse
35
6. Solution of Linear Equations
37
7. Linear Combinations, Span, Linear Independence
39
8. Basis
41
9. Rank
43
10. Solving Systems with More Variables than Equations
45
11. Solving Linear Programs with Matlab
47
Chapter 4. Convex Sets, Functions and Cones and Polyhedral Theory
51
1. Convex Sets
51
2. Convex and Concave Functions
52
3. Polyhedral Sets
53
4. Rays and Directions
56
5. Directions of Polyhedral Sets
57
6. Extreme Points
59
7. Extreme Directions
62
iii
8. Caratheodory Characterization Theorem
64
Chapter 5. The Simplex Method
69
1. Linear Programming and Extreme Points
69
2. Algorithmic Characterization of Extreme Points
70
3. The Simplex Algorithm?Algebraic Form
71
4. Simplex Method?Tableau Form
78
5. Identifying Unboundedness
81
6. Identifying Alternative Optimal Solutions
84
7. Degeneracy and Convergence
86
Chapter 6. Simplex Initialization
91
1. Artificial Variables
91
2. The Two-Phase Simplex Algorithm
95
3. The Big-M Method
98
4. The Single Artificial Variable Technique
102
5. Problems that Can't be Initialized by Hand
103
Chapter 7. Degeneracy and Convergence
109
1. Degeneracy Revisited
109
2. The Lexicographic Minimum Ratio Leaving Variable Rule
111
3. Bland's Rule, Entering Variable Rules and Other Considerations
116
Chapter 8. The Revised Simplex Method and Optimality Conditions
117
1. The Revised Simplex Method
117
2. Farkas' Lemma and Theorems of the Alternative
121
3. The Karush-Kuhn-Tucker Conditions
126
4. Relating the KKT Conditions to the Tableau
132
Chapter 9. Duality
137
1. The Dual Problem
137
2. Weak Duality
141
3. Strong Duality
142
4. Geometry of the Dual Problem
145
5. Economic Interpretation of the Dual Problem
148
6. The Dual Simplex Method
152
Bibliography
157
iv
List of Figures
1.1 Goat pen with unknown side lengths. The objective is to identify the values of
x and y that maximize the area of the pen (and thus the number of goats that
can be kept).
2
1.2 Plot with Level Sets Projected on the Graph of z. The level sets existing in R2
while the graph of z existing R3. The level sets have been projected onto their
appropriate heights on the graph.
5
1.3 Contour Plot of z = x2 + y2. The circles in R2 are the level sets of the function. The lighter the circle hue, the higher the value of c that defines the level set. 6
1.4 A Line Function: The points in the graph shown in this figure are in the set
produced using the expression x0 + vt where x0 = (2, 1) and let v = (2, 2).
6
1.5 A Level Curve Plot with Gradient Vector: We've scaled the gradient vector
in this case to make the picture understandable. Note that the gradient is
perpendicular to the level set curve at the point (1, 1), where the gradient was
evaluated. You can also note that the gradient is pointing in the direction of
steepest ascent of z(x, y).
8
1.6 Level Curves and Feasible Region: At optimality the level curve of the objective
function is tangent to the binding constraints.
11
1.7 Gradients of the Binding Constraint and Objective: At optimality the gradient
of the binding constraints and the objective function are scaled versions of each
other.
12
2.1 Feasible Region and Level Curves of the Objective Function: The shaded region
in the plot is the feasible region and represents the intersection of the five
inequalities constraining the values of x1 and x2. On the right, we see the optimal solution is the "last" point in the feasible region that intersects a level
set as we move in the direction of increasing profit.
16
2.2 A Bounded Set: The set S (in blue) is bounded because it can be entirely
contained inside a ball of a finite radius r and centered at some point x0. In this example, the set S is in R2. This figure also illustrates the fact that a ball in R2
is just a disk and its boundary.
18
2.3 An example of infinitely many alternative optimal solutions in a linear programming problem. The level curves for z(x1, x2) = 18x1 + 6x2 are parallel to one face of the polygon boundary of the feasible region. Moreover, this side contains the points of greatest value for z(x1, x2) inside the feasible region. Any
v
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