This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Lecture 12: Necessary and Sufficient Conditions for Optimality of PrimalDual Pairs February 28, 2007 Lecture 12 Outline • General Form of Dual Problem • Necessary and Sufficient Optimality Condition for PrimalDual Pairs • KarushKuhnTucker (KKT) Conditions • Examples Convex Optimization 1 Lecture 12 General Convex Problem and Its Dual Primal Problem minimize f ( x ) subject to g j ( x ) ≤ , j = 1 ,...,m a T i x = b i , i = 1 ,...,r x ∈ X When equalities are kept, we have: Lagrangian Function L ( x,μ,λ ) = f ( x )+ μ T g ( x )+ λ T ( Ax b ) , μ ∈ R m , μ , λ ∈ R r where g = ( g 1 ,...,g m ) T and A is a matrix with rows a T i , i = 1 ,...,r Dual Function q ( μ,λ ) = inf x ∈ X L ( x,μ,λ ) = inf x ∈ X n f ( x ) + μ T g ( x ) + λ T ( Ax b ) o The infimum is actually taken over X ∩ dom f ∩ dom g 1 ∩ ... ∩ dom g m Dual Problem max μ ≥ , λ ∈ R r q ( μ,λ ) Convex Optimization 2 Lecture 12 Optimality Condition for PrimalDual Pairs Theorem Consider convex primal problem with finite optimal value f * . Assume there is no duality gap , i.e., q * = f * . Then: x * is a primal optimal and ( μ * ,λ * ) is a dual optimal if and only if • Primal Feasibility : x * primal feasible i.e., g ( x * ) ≤ , Ax * = b, x * ∈ X ∩ dom f • Dual Feasibility : ( μ * ,λ * ) is dual feasible i.e., μ * • Lagrangian Optimality in x : x * attains the minimum in inf x ∈ X L ( x,μ * ,λ * ) • Lagrangian Optimality in ( μ,λ ) : ( μ * ,λ * ) attains the maximum in sup μ ≥ , λ ∈ R r L ( x * ,μ,λ ) Convex Optimization 3 Lecture 12 Proof ( ⇒ ) Let x * be primal optimal and ( μ * ,λ * ) be dual optimal. Then, they are feasible for primal and dual problems, respectively. By the optimality of these vectors, we have f * = f ( x * ) and q * = q ( μ * ,λ * ) . By the no gap relation f * = q * , we obtain f ( x * ) = q ( μ * ,λ * ) = inf x ∈ X n f ( x ) + ( μ * ) T g ( x ) + ( λ * ) T ( Ax b ) o ≤ f ( x * ) + ( μ * ) T g ( x * ) + ( λ * ) T ( Ax * b ) ≤ f ( x * ) Hence, the inequalities must hold as equalities, implying that inf x ∈ X L ( x,μ * ,λ * ) = inf x ∈ X n f ( x ) + ( μ * ) T g ( x ) + ( λ * ) T ( Ax b ) o = f ( x * ) + ( μ * ) T g ( x * ) + ( λ * ) T ( Ax * b ) = L ( x * ,μ * ,λ * ) Thus, x * attains the minimum in inf x ∈ X L ( x,μ * ,λ * ) Convex Optimization 4 Lecture 12 Also, it follows that q ( μ * ,λ * ) = f ( x * ) + ( μ * ) T g ( x * ) + ( λ * ) T ( Ax * b ) = L ( x * ,μ * ,λ * ) (1) Furthermore, we have f ( x * ) = q ( μ * ,λ * ) = sup μ ≥ ,λ ∈ R r q ( μ,λ ) = sup μ ≥ ,λ ∈ R r inf...
View
Full Document
 Spring '07
 AngeliaNedich
 Optimization, Mathematical optimization, Dual problem, Convex Optimization

Click to edit the document details