Approximate Farkas Lemmas in Convex Optimization

Transcription

1 Approximate Farkas Lemmas in Convex Optimization Imre McMaster University Advanced Optimization Lab AdvOL Graduate Student Seminar October 25, 2004

2 1 Exact Farkas Lemma Motivation 2 3 Future plans

3 The Farkas Lemma Outline Exact Farkas Lemma Motivation The following are equivalent x : Ax = b (1) x 0. (2) y : A T y 0 (3) b T y = 1. (4) Certificate for infeasibility in a perfect world... Almost certificate?

4 The Farkas Lemma Outline Exact Farkas Lemma Motivation The following are equivalent x : Ax = b (1) x 0. (2) y : A T y 0 (3) b T y = 1. (4) Certificate for infeasibility in a perfect world... Almost certificate?

5 Why approximate? Outline Exact Farkas Lemma Motivation Practical infeasibility Numerical accuracy Natural bounds Stopping criteria Advanced infeasibility detection Sensitivity analysis

6 Why approximate? Outline Exact Farkas Lemma Motivation Practical infeasibility Numerical accuracy Natural bounds Stopping criteria Advanced infeasibility detection Sensitivity analysis

7 Why approximate? Outline Exact Farkas Lemma Motivation Practical infeasibility Numerical accuracy Natural bounds Stopping criteria Advanced infeasibility detection Sensitivity analysis

8 Why approximate? Outline Exact Farkas Lemma Motivation Practical infeasibility Numerical accuracy Natural bounds Stopping criteria Advanced infeasibility detection Sensitivity analysis

9 Outline α x = min x Ax = b x 0 Theorem α x β u = 1 ( 0 = 1 ) Proof. Easy, both are linear systems. β u = min u 1 A T y u b T y = 1

10 Overview of conic duality The cone: K R n closed, convex, pointed, nonempty interior Dual cone: K = { s R n : x T s 0, x K } Ordering: x K 0 x K Primal problem Ax = b (5) x K 0. (6) Dual problem A T y K 0 (7) b T y = 1. (8) Primal is solvable Dual is not solvable Primal is not solvable Dual is almost solvable

11 Overview of conic duality The cone: K R n closed, convex, pointed, nonempty interior Dual cone: K = { s R n : x T s 0, x K } Ordering: x K 0 x K Primal problem Ax = b (5) x K 0. (6) Dual problem A T y K 0 (7) b T y = 1. (8) Primal is solvable Dual is not solvable Primal is not solvable Dual is almost solvable

12 Overview of conic duality The cone: K R n closed, convex, pointed, nonempty interior Dual cone: K = { s R n : x T s 0, x K } Ordering: x K 0 x K Primal problem Ax = b (5) x K 0. (6) Dual problem A T y K 0 (7) b T y = 1. (8) Primal is solvable Dual is not solvable Primal is not solvable Dual is almost solvable

13 Overview of conic duality The cone: K R n closed, convex, pointed, nonempty interior Dual cone: K = { s R n : x T s 0, x K } Ordering: x K 0 x K Primal problem Ax = b (5) x K 0. (6) Dual problem A T y K 0 (7) b T y = 1. (8) Primal is solvable Dual is not solvable Primal is not solvable Dual is almost solvable

14 Approximate Farkas Lemma for CO Theorem α x β u = 1 [ 0 = 1 ] α x = min x Ax = b x K 0 β u = min u 1 A T y K u b T y = 1 Proof. More complicated.

15 Proof of the Approximate Farkas Lemma for CO Perturbed system: αx ε := min x Ax = b ε x K v ε b b ε ε v ε ε. α ε x α x (ε 0) If the original is feasible then α ε x and α x are realized The rest is conic duality

16 Proof of the Approximate Farkas Lemma for CO Perturbed system: αx ε := min x Ax = b ε x K v ε b b ε ε v ε ε. α ε x α x (ε 0) If the original is feasible then α ε x and α x are realized The rest is conic duality

17 Proof of the Approximate Farkas Lemma for CO Perturbed system: αx ε := min x Ax = b ε x K v ε b b ε ε v ε ε. α ε x α x (ε 0) If the original is feasible then α ε x and α x are realized The rest is conic duality

18 Proof of the Approximate Farkas Lemma for CO Perturbed system: αx ε := min x Ax = b ε x K v ε b b ε ε v ε ε. α ε x α x (ε 0) If the original is feasible then α ε x and α x are realized The rest is conic duality

19 Future plans Outline Future plans Work in progress! Derive stopping criteria for CO infeasible and embedding methods Prove complexity Implement (McIPM, SeDuMi(!)) Tests Generalize for Convex Optimization

20 Future plans Outline Future plans Work in progress! Derive stopping criteria for CO infeasible and embedding methods Prove complexity Implement (McIPM, SeDuMi(!)) Tests Generalize for Convex Optimization

21 Future plans Outline Future plans Work in progress! Derive stopping criteria for CO infeasible and embedding methods Prove complexity Implement (McIPM, SeDuMi(!)) Tests Generalize for Convex Optimization