v2007.11.26 - Convex Optimization

4.2. FRAMEWORK 239Optimal value of the dual objective thus represents the greatest lower boundon the primal. This fact is known as the weak duality theorem for semidefiniteprogramming, [301,1.3.8] and can be used to detect convergence in anyprimal/dual numerical method of solution.4.2.3 Optimality conditionsWhen any primal feasible point exists relatively interior to A ∩ S n + in S n ,or when any dual feasible point exists relatively interior to C ∗ in S n × R m ,then by Slater’s sufficient condition these two problems (547P) and (547D)become strong duals. In other words, the primal optimal objective valuebecomes equivalent to the dual optimal objective value: there is no dualitygap (Figure 47); id est, if ∃X ∈ A ∩ int S n + or ∃S,y ∈ rel int C ∗ then〈C , X ⋆ 〉 = 〈b, y ⋆ 〉〈 ∑iy ⋆ i A i + S ⋆ , X ⋆ 〉= [ 〈A 1 , X ⋆ 〉 · · · 〈A m , X ⋆ 〉 ] y ⋆〈S ⋆ , X ⋆ 〉 = 0(571)where S ⋆ , y ⋆ denote a dual optimal solution. 4.9 We summarize this:4.2.3.0.1 Corollary. Optimality and strong duality. [271,3.1][301,1.3.8] For semidefinite programs (547P) and (547D), assume primaland dual feasible sets A ∩ S n + ⊂ S n and C ∗ ⊂ S n × R m (559) are nonempty.ThenX ⋆ is optimal for (P)S ⋆ , y ⋆ are optimal for (D)the duality gap 〈C,X ⋆ 〉−〈b, y ⋆ 〉 is 0if and only ifi) ∃X ∈ A ∩ int S n + or ∃S , y ∈ rel int C ∗andii) 〈S ⋆ , X ⋆ 〉 = 0⋄4.9 Optimality condition 〈S ⋆ , X ⋆ 〉=0 is called a complementary slackness condition, inkeeping with the tradition of linear programming, [64] that forbids dual inequalities in(547) to simultaneously hold strictly. [231,4]