Lecture 2 Piecewise-linear optimization
[Pages:24]L. Vandenberghe
EE236A (Fall 2013-14)
Lecture 2 Piecewise-linear optimization
? piecewise-linear minimization ? 1- and -norm approximation ? examples ? modeling software
2?1
Linear and affine functions
linear function: a function f : Rn R is linear if f (x + y) = f (x) + f (y) x, y Rn, , R
property: f is linear if and only if f (x) = aT x for some a
affine function: a function f : Rn R is affine if f (x + (1 - )y) = f (x) + (1 - )f (y) x, y Rn, R
property: f is affine if and only if f (x) = aT x + b for some a, b
Piecewise-linear optimization
2?2
Piecewise-linear function
f : Rn R is (convex) piecewise-linear if it can be expressed as f (x) = i=m1,a..x.,m(aTi x + bi)
f is parameterized by m n-vectors ai and m scalars bi
f (x)
aTi x + bi x
(the term piecewise-affine is more accurate but less common)
Piecewise-linear optimization
2?3
Piecewise-linear minimization
minimize f (x) = i=m1,a..x.,m(aTi x + bi)
? equivalent LP (with variables x and auxiliary scalar variable t) minimize t subject to aTi x + bi t, i = 1, . . . , m
to see equivalence, note that for fixed x the optimal t is t = f (x)
? LP in matrix notation: minimize c~T x~ subject to A~x~ ~b with
x~ =
x t
,
c~ =
0 1
,
A~ =
a..T1
-1 .. ,
aTm -1
~b =
-..b1
-bm
Piecewise-linear optimization
2?4
Minimizing a sum of piecewise-linear functions
minimize f (x) + g(x) = i=m1,a..x.,m(aTi x + bi) + i=m1,a..x.,p(ciT x + di) ? cost function is piecewise-linear: maximum of mp affine functions
f (x) + g(x) = max (ai + cj)T x + (bi + dj) i=1,...,m j=1,...,p
? equivalent LP with m + p inequalities
minimize t1 + t2 subject to aTi x + bi t1,
cTi x + di t2,
i = 1, . . . , m i = 1, . . . , p
note that for fixed x, optimal t1, t2 are t1 = f (x), t2 = g(x)
Piecewise-linear optimization
2?5
? equivalent LP in matrix notation
minimize c~T x~ subject to A~x~ ~b
with
x x~ = t1 ,
t2
0 c~ = 1 ,
1
a..T1
A~ =
aTm c..T1
cTp
-1 0 .. ..
-1 0
0 -1
,
..
..
0 -1
-..b1
~b
=
-bm -..d1
-dp
Piecewise-linear optimization
2?6
-Norm (Cheybshev) approximation
minimize Ax - b with A Rm?n, b Rm
? -norm (Chebyshev norm) of m-vector y is
y
=
max
i=1,...,m
|yi|
=
max
i=1,...,m
max{yi,
-yi}
? equivalent LP (with variables x and auxiliary scalar variable t)
minimize t subject to -t1 Ax - b t1
(for fixed x, optimal t is t = Ax - b )
Piecewise-linear optimization
2?7
? equivalent LP in matrix notation
minimize subject to
0T x
1
t
A -1 -A -1
x t
b -b
Piecewise-linear optimization
2?8
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