Optimization under uncertainty: robust optimization

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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

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

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

CES 735/8 Imre Pólik Robust linear optimization Outline Motivation Theoretical background Robust linear optimization Robust least squares optimization Applications

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

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

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

CES 735/8 Imre Pólik Robust least squares optimization Outline Motivation Theoretical background Robust linear optimization Robust least squares optimization Applications

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 2 2 12 10 8 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 2 2 2 figure shows r(u) = A(u)x b 2 0 2 1 0 1 2 u Approximation and fitting 6 17

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

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 2 2 1 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

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 0.25 0.2 x rls frequency 0.15 0.1 0.05 x tik x ls 0 0 1 2 3 4 5 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

CES 735/8 Imre Pólik Truss topology design Outline Motivation Theoretical background Applications Truss topology design Antenna array design

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CES 735/8 Imre Pólik Antenna array design Outline Motivation Theoretical background Applications Truss topology design Antenna array design

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