v2010.10.26 - Convex Optimization

4.2. FRAMEWORK 2894.2.2.1 Dual problem statement is not uniqueEven subtle but equivalent restatements of a primal convex problem canlead to vastly different statements of a corresponding dual problem. Thisphenomenon is of interest because a particular instantiation of dual problemmight be easier to solve numerically or it might take one of few forms forwhich analytical solution is known.Here is a canonical restatement of prototypical dual semidefinite program(649D), for example, equivalent by (194):(D)maximizey∈R m , S∈S n 〈b, y〉subject to S ≽ 0svec −1 (A T y) + S = C≡maximizey∈R m 〈b, y〉subject to svec −1 (A T y) ≼ C(649˜D)Dual feasible cone interior in int S n + (661) (651) thereby corresponds withcanonical dual (˜D) feasible interior{}rel int ˜Dm∑∗ y ∈ R m | y i A i ≺ C(673)i=14.2.2.1.1 Exercise. Primal prototypical semidefinite program.Derive prototypical primal (649P) from its canonical dual (649˜D); id est,demonstrate that particular connectivity in Figure 82. 4.2.3 Optimality conditionsWhen primal feasible cone interior A ∩ int S n + exists in S n or when canonicaldual feasible interior rel int ˜D ∗ exists in R m , then these two problems (649P)(649D) become strong duals by Slater’s sufficient condition (p.285). In otherwords, the primal optimal objective value becomes equal to the dual optimalobjective value: there is no duality gap (Figure 58) and so determination ofconvergence is facilitated; id est, if ∃X ∈ A ∩ int S n + or ∃y ∈ rel int ˜D ∗ then〈 ∑i〈C , X ⋆ 〉 = 〈b, y ⋆ 〉y ⋆ i A i + S ⋆ , X ⋆ 〉= [ 〈A 1 , X ⋆ 〉 · · · 〈A m , X ⋆ 〉 ] y ⋆〈S ⋆ , X ⋆ 〉 = 0(674)