approximate farkas lemmas in convex optimization
TRANSCRIPT
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