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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...
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This note was uploaded on 08/22/2008 for the course GE 498 AN taught by Professor Angelianedich during the Spring '07 term at University of Illinois at Urbana–Champaign.
 Spring '07
 AngeliaNedich

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