approximate farkas lemmas in convex optimization

OutlineIntroduction

ResultsClosing

Approximate Farkas Lemmas inConvex Optimization

Imre Polik

McMaster UniversityAdvanced Optimization Lab

AdvOL Graduate Student SeminarOctober 25, 2004

Polik Approximate Farkas Lemmas in Convex Optimization

OutlineIntroduction

ResultsClosing

1 IntroductionExact Farkas LemmaMotivation

2 ResultsLinear optimizationConic optimization

3 ClosingFuture plans

Polik Approximate Farkas Lemmas in Convex Optimization

OutlineIntroduction

ResultsClosing

Exact Farkas LemmaMotivation

The Farkas Lemma

The following are equivalent

∃x : Ax = b (1)

x ≥ 0. (2)

@y : AT y ≤ 0 (3)

bT y = 1. (4)

Certificate for infeasibility in a perfect world...

Almost certificate?

Polik Approximate Farkas Lemmas in Convex Optimization

OutlineIntroduction

ResultsClosing

Exact Farkas LemmaMotivation

The Farkas Lemma

The following are equivalent

∃x : Ax = b (1)

x ≥ 0. (2)

@y : AT y ≤ 0 (3)

bT y = 1. (4)

Certificate for infeasibility in a perfect world...

Almost certificate?

Polik Approximate Farkas Lemmas in Convex Optimization

OutlineIntroduction

ResultsClosing

Exact Farkas LemmaMotivation

Why approximate?

Practical infeasibility

Numerical accuracyNatural bounds

Stopping criteria

Advanced infeasibility detection

Sensitivity analysis

Polik Approximate Farkas Lemmas in Convex Optimization

OutlineIntroduction

ResultsClosing

Exact Farkas LemmaMotivation

Why approximate?

Practical infeasibility

Numerical accuracyNatural bounds

Stopping criteria

Advanced infeasibility detection

Sensitivity analysis

Polik Approximate Farkas Lemmas in Convex Optimization

OutlineIntroduction

ResultsClosing

Exact Farkas LemmaMotivation

Why approximate?

Practical infeasibility

Numerical accuracyNatural bounds

Stopping criteria

Advanced infeasibility detection

Sensitivity analysis

Polik Approximate Farkas Lemmas in Convex Optimization

OutlineIntroduction

ResultsClosing

Exact Farkas LemmaMotivation

Why approximate?

Practical infeasibility

Numerical accuracyNatural bounds

Stopping criteria

Advanced infeasibility detection

Sensitivity analysis

Polik Approximate Farkas Lemmas in Convex Optimization

OutlineIntroduction

ResultsClosing

Linear optimizationConic optimization

Linear optimization

αx = min ‖x‖∞Ax = b

x ≥ 0

βu = min ‖u‖1AT y ≤ u

bT y = 1

Theorem

αxβu = 1 (”0 · ∞ = 1”)

Proof.

Easy, both are linear systems.Polik Approximate Farkas Lemmas in Convex Optimization

OutlineIntroduction

ResultsClosing

Linear optimizationConic optimization

Overview of conic duality

The cone: K ⊂ Rn closed, convex, pointed, nonempty interior

Dual cone: K∗ ={s ∈ Rn : xT s ≥ 0,∀x ∈ K

}Ordering: x �K 0 ⇔ x ∈ KPrimal problem

Ax = b (5)

x �K 0. (6)

Dual problemAT y �K∗ 0 (7)

bT y = 1. (8)

Primal is solvable ⇒ Dual is not solvable

Primal is not solvable ⇒ Dual is almost solvable

Polik Approximate Farkas Lemmas in Convex Optimization

OutlineIntroduction

ResultsClosing

Linear optimizationConic optimization

Overview of conic duality

The cone: K ⊂ Rn closed, convex, pointed, nonempty interior

Dual cone: K∗ ={s ∈ Rn : xT s ≥ 0,∀x ∈ K

}Ordering: x �K 0 ⇔ x ∈ KPrimal problem

Ax = b (5)

x �K 0. (6)

Dual problemAT y �K∗ 0 (7)

bT y = 1. (8)

Primal is solvable ⇒ Dual is not solvable

Primal is not solvable ⇒ Dual is almost solvable

Polik Approximate Farkas Lemmas in Convex Optimization

OutlineIntroduction

ResultsClosing

Linear optimizationConic optimization

Overview of conic duality

The cone: K ⊂ Rn closed, convex, pointed, nonempty interior

Dual cone: K∗ ={s ∈ Rn : xT s ≥ 0,∀x ∈ K

}Ordering: x �K 0 ⇔ x ∈ KPrimal problem

Ax = b (5)

x �K 0. (6)

Dual problemAT y �K∗ 0 (7)

bT y = 1. (8)

Primal is solvable ⇒ Dual is not solvable

Primal is not solvable ⇒ Dual is almost solvable

Polik Approximate Farkas Lemmas in Convex Optimization

OutlineIntroduction

ResultsClosing

Linear optimizationConic optimization

Overview of conic duality

The cone: K ⊂ Rn closed, convex, pointed, nonempty interior

Dual cone: K∗ ={s ∈ Rn : xT s ≥ 0,∀x ∈ K

}Ordering: x �K 0 ⇔ x ∈ KPrimal problem

Ax = b (5)

x �K 0. (6)

Dual problemAT y �K∗ 0 (7)

bT y = 1. (8)

Primal is solvable ⇒ Dual is not solvable

Primal is not solvable ⇒ Dual is almost solvable

Polik Approximate Farkas Lemmas in Convex Optimization

OutlineIntroduction

ResultsClosing

Linear optimizationConic optimization

Approximate Farkas Lemma for CO

αx = min ‖x‖∞Ax = b

x �K 0

βu = min ‖u‖1AT y �K∗ u

bT y = 1

Theorem

αxβu = 1 [”0 · ∞ = 1”]

Proof.

More complicated.Polik Approximate Farkas Lemmas in Convex Optimization

OutlineIntroduction

ResultsClosing

Linear optimizationConic optimization

Proof of the Approximate Farkas Lemma for CO

Perturbed system:αε

x := min ‖x‖∞Ax = bε

x �K vε

‖b− bε‖∞ ≤ ε

‖vε‖∞ ≤ ε.

αεx → αx (ε → 0)

If the original is feasible then αεx and αx are realized

The rest is conic duality

Polik Approximate Farkas Lemmas in Convex Optimization

OutlineIntroduction

ResultsClosing

Linear optimizationConic optimization

Proof of the Approximate Farkas Lemma for CO

Perturbed system:αε

x := min ‖x‖∞Ax = bε

x �K vε

‖b− bε‖∞ ≤ ε

‖vε‖∞ ≤ ε.

αεx → αx (ε → 0)

If the original is feasible then αεx and αx are realized

The rest is conic duality

Polik Approximate Farkas Lemmas in Convex Optimization

OutlineIntroduction

ResultsClosing

Linear optimizationConic optimization

Proof of the Approximate Farkas Lemma for CO

Perturbed system:αε

x := min ‖x‖∞Ax = bε

x �K vε

‖b− bε‖∞ ≤ ε

‖vε‖∞ ≤ ε.

αεx → αx (ε → 0)

If the original is feasible then αεx and αx are realized

The rest is conic duality

Polik Approximate Farkas Lemmas in Convex Optimization

OutlineIntroduction

ResultsClosing

Linear optimizationConic optimization

Proof of the Approximate Farkas Lemma for CO

Perturbed system:αε

x := min ‖x‖∞Ax = bε

x �K vε

‖b− bε‖∞ ≤ ε

‖vε‖∞ ≤ ε.

αεx → αx (ε → 0)

If the original is feasible then αεx and αx are realized

The rest is conic duality

Polik Approximate Farkas Lemmas in Convex Optimization

OutlineIntroduction

ResultsClosing

Future plans

Future plans

Work in progress!

Derive stopping criteria for CO

infeasible and embedding methodsProve complexityImplement (McIPM, SeDuMi(!))Tests

Generalize for Convex Optimization

Polik Approximate Farkas Lemmas in Convex Optimization

OutlineIntroduction

ResultsClosing

Future plans

Future plans

Work in progress!

Derive stopping criteria for CO

infeasible and embedding methodsProve complexityImplement (McIPM, SeDuMi(!))Tests

Generalize for Convex Optimization

Polik Approximate Farkas Lemmas in Convex Optimization

OutlineIntroduction

ResultsClosing

Future plans

Future plans

Work in progress!

Derive stopping criteria for CO

infeasible and embedding methodsProve complexityImplement (McIPM, SeDuMi(!))Tests

Generalize for Convex Optimization

Polik Approximate Farkas Lemmas in Convex Optimization