v2006.03.09 - Convex Optimization

6.2. FRAMEWORK 3676.2.2 DualsThe dual objective function evaluated at any feasible point represents a lowerbound on the primal optimal objective value. We can see this by directsubstitution: Assume the feasible sets A ∩ S n + and C ∗ are nonempty. Thenit is always true:〈 ∑i〈C , X〉 ≥ 〈b, y〉〉y i A i + S , X ≥ [ 〈A 1 , X〉 · · · 〈A m , X〉 ] y〈S , X〉 ≥ 0(867)The converse also follows becauseX ≽ 0, S ≽ 0 ⇒ 〈S,X〉 ≥ 0 (1082)The optimal value of the dual objective thus represents the greatest lowerbound on the primal. This fact is known as the weak duality theoremfor semidefinite programming, [259,1.3.8] and can be used to detectconvergence in any primal/dual numerical method of solution.6.2.3 Optimality conditionsWhen any primal feasible point exists relatively interior to A ∩ S n + in S n , orwhen any dual feasible point exists relatively interior to C ∗ in S n × R m , thenby Slater’s sufficient condition these two problems (845)(P) and (845)(D)become strong duals. In other words, the primal optimal objective valuebecomes equivalent to the dual optimal objective value: there is no dualitygap; id est, if ∃X ∈ A ∩ int S n + or ∃S,y ∈ rel int C ∗ then〈 ∑i〈C , X ⋆ 〉 = 〈b, y ⋆ 〉y ⋆ i A i + S ⋆ , X ⋆ 〉= [ 〈A 1 , X ⋆ 〉 · · · 〈A m , X ⋆ 〉 ] y ⋆〈S ⋆ , X ⋆ 〉 = 0(868)where S ⋆ , y ⋆ denote a dual optimal solution. 6.9 We summarize this:6.9 Optimality condition 〈S ⋆ , X ⋆ 〉=0 is called a complementary slackness condition, inkeeping with the tradition of linear programming, [53] that forbids dual inequalities in(845) to simultaneously hold strictly. [193,4]