v2010.10.26 - Convex Optimization

290 CHAPTER 4. SEMIDEFINITE PROGRAMMINGP˜PdualitydualityDtransformation˜DFigure 82: Connectivity indicates paths between particular primal and dualproblems from Exercise 4.2.2.1.1. More generally, any path between primalproblems P (and equivalent ˜P) and dual D (and equivalent ˜D) is possible:implying, any given path is not necessarily circuital; dual of a dual problemis not necessarily stated in precisely same manner as corresponding primalconvex problem, in other words, although its solution set is equivalent towithin some transformation.where S ⋆ , y ⋆ denote a dual optimal solution. 4.13 We summarize this:4.2.3.0.1 Corollary. Optimality and strong duality. [359,3.1][391,1.3.8] For semidefinite programs (649P) and (649D), assume primaland dual feasible sets A ∩ S n + ⊂ S n and D ∗ ⊂ S n × R m (661) are nonempty.ThenX ⋆ is optimal for (649P)S ⋆ , y ⋆ are optimal for (649D)duality gap 〈C,X ⋆ 〉−〈b, y ⋆ 〉 is 0if and only ifi) ∃X ∈ A ∩ int S n + or ∃y ∈ rel int ˜D ∗ii) 〈S ⋆ , X ⋆ 〉 = 0and⋄4.13 Optimality condition 〈S ⋆ , X ⋆ 〉=0 is called a complementary slackness condition, inkeeping with linear programming tradition [94], that forbids dual inequalities in (649) tosimultaneously hold strictly. [306,4]