Optimization under uncertainty: robust optimization
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- Ανάργυρος Φραγκούδης
- 6 χρόνια πριν
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1 CES 735/8 Imre Pólik Outline Motivation Theoretical background Applications Optimization under uncertainty: robust optimization Imre Pólik, PhD McMaster University School of Computational Engineering and Science March 6, 2008
2 CES 735/8 Imre Pólik Outline Outline Motivation Theoretical background 1 Motivation The role of uncertainty Applications 2 Theoretical background Robust linear optimization Robust least squares optimization 3 Applications Truss topology design Antenna array design
3 Outline CES 735/8 Imre Pólik Motivation The role of uncertainty Theoretical background Applications Why uncertainty? Design on computer, measure with a ruler, cut with an axe! Uncertain data measurement errors noise unpredictability vibration Implementation errors technological limits imperfect materials
4 CES 735/8 Imre Pólik Robust linear optimization Outline Motivation Theoretical background Robust linear optimization Robust least squares optimization Applications
5 Robust linear programming the parameters in optimization problems are often uncertain, e.g., in an LP there can be uncertainty in c, a i, b i minimize c T x subject to a T i x b i, i = 1,..., m, two common approaches to handling uncertainty (in a i, for simplicity) deterministic model: constraints must hold for all a i E i minimize c T x subject to a T i x b i for all a i E i, i = 1,..., m, stochastic model: a i is random variable; constraints must hold with probability η minimize c T x subject to prob(a T i x b i) η, i = 1,..., m Convex optimization problems 4 26
6 deterministic approach via SOCP choose an ellipsoid as E i : E i = {ā i P i u u 2 1} (ā i R n, P i R n n ) center is ā i, semi-axes determined by singular values/vectors of P i robust LP minimize c T x subject to a T i x b i a i E i, i = 1,..., m is equivalent to the SOCP minimize c T x subject to ā T i x P i T x 2 b i, i = 1,..., m (follows from sup u 2 1(ā i P i u) T x = ā T i x P i T x 2) Convex optimization problems 4 27
7 stochastic approach via SOCP assume a i is Gaussian with mean ā i, covariance Σ i (a i N (ā i, Σ i )) a T i x is Gaussian r.v. with mean āt i x, variance xt Σ i x; hence ( ) prob(a T b i ā T i i x b i ) = Φ x Σ 1/2 i x 2 where Φ(x) = (1/ 2π) x e t2 /2 dt is CDF of N (0, 1) robust LP minimize c T x subject to prob(a T i x b i) η, i = 1,..., m, with η 1/2, is equivalent to the SOCP minimize c T x subject to ā T i x Φ 1 (η) Σ 1/2 i x 2 b i, i = 1,..., m Convex optimization problems 4 28
8 CES 735/8 Imre Pólik Robust least squares optimization Outline Motivation Theoretical background Robust linear optimization Robust least squares optimization Applications
9 minimize Ax b with uncertain A Robust approximation two approaches: PSfrag replacements stochastic: assume A is random, minimize E Ax b worst-case: set A of possible values of A, minimize sup A A Ax b tractable only in special cases (certain norms, distributions, sets A) example: A(u) = A 0 ua 1 x nom minimizes A 0 x b x nom x stoch minimizes E A(u)x b 2 2 with u uniform on [ 1, 1] r(u) 6 4 x stoch x wc x wc minimizes sup 1 u 1 A(u)x b figure shows r(u) = A(u)x b u Approximation and fitting 6 17
10 stochastic robust LS with A = Ā U, U random, E U = 0, E U T U = P minimize E (Ā U)x b 2 2 explicit expression for objective: E Ax b 2 2 = E Āx b Ux 2 2 = Āx b 2 2 E x T U T Ux = Āx b 2 2 x T P x hence, robust LS problem is equivalent to LS problem minimize Āx b 2 2 P 1/2 x 2 2 for P = δi, get Tikhonov regularized problem minimize Āx b 2 2 δ x 2 2 Approximation and fitting 6 18
11 worst-case robust LS with A = {Ā u 1A 1 u p A p u 2 1} minimize sup A A Ax b 2 2 = sup u 2 1 P (x)u q(x) 2 2 where P (x) = [ A 1 x A 2 x A p x ], q(x) = Āx b from page 5 14, strong duality holds between the following problems maximize P u q 2 2 subject to u minimize subject to t λ I P q P T λi 0 q T 0 t 0 hence, robust LS problem is equivalent to SDP minimize subject to t λ I P (x) q(x) P (x) T λi 0 q(x) T 0 t 0 Approximation and fitting 6 19
12 PSfrag replacements example: histogram of residuals r(u) = (A 0 u 1 A 1 u 2 A 2 )x b 2 with u uniformly distributed on unit disk, for three values of x x rls frequency x tik x ls r(u) x ls minimizes A 0 x b 2 x tik minimizes A 0 x b 2 2 x 2 2 (Tikhonov solution) x wc minimizes sup u 2 1 A 0 x b 2 2 x 2 2 Approximation and fitting 6 20
13 CES 735/8 Imre Pólik Truss topology design Outline Motivation Theoretical background Applications Truss topology design Antenna array design
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21 CES 735/8 Imre Pólik Antenna array design Outline Motivation Theoretical background Applications Truss topology design Antenna array design
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