v2010.10.26 - Convex Optimization

282 CHAPTER 4. SEMIDEFINITE PROGRAMMING4.1.2.2.1 Example. Optimization over A ∩ S 3 + .Consider minimization of the real linear function 〈C , X〉 overa polyhedral feasible set;P A ∩ S 3 + (654)f0 ⋆ minimize 〈C , X〉Xsubject to X ∈ A ∩ S+3(655)As illustrated for particular vector C and hyperplane A = ∂H in Figure 81,this linear function is minimized on any X belonging to the face of Pcontaining extreme points {Γ 1 , Γ 2 } and all the rank-2 matrices in between;id est, on any X belonging to the face of PF(P) = {X | 〈C , X〉 = f ⋆ 0 } ∩ A ∩ S 3 + (656)exposed by the hyperplane {X | 〈C , X〉=f ⋆ 0 }. In other words, the set of alloptimal points X ⋆ is a face of P{X ⋆ } = F(P) = Γ 1 Γ 2 (657)comprising rank-1 and rank-2 positive semidefinite matrices. Rank 1 isthe upper bound on existence in the feasible set P for this case m = 1hyperplane constituting A . The rank-1 matrices Γ 1 and Γ 2 in face F(P)are extreme points of that face and (by transitivity (2.6.1.2)) extremepoints of the intersection P as well. As predicted by analogy to Barvinok’sProposition 2.9.3.0.1, the upper bound on rank of X existent in the feasibleset P is satisfied by an extreme point. The upper bound on rank of anoptimal solution X ⋆ existent in F(P) is thereby also satisfied by an extremepoint of P precisely because {X ⋆ } constitutes F(P) ; 4.7 in particular,{X ⋆ ∈ P | rankX ⋆ ≤ 1} = {Γ 1 , Γ 2 } ⊆ F(P) (658)As all linear functions on a polyhedron are minimized on a face, [94] [252][274] [280] by analogy we so demonstrate coexistence of optimal solutions X ⋆of (649P) having assorted rank.4.7 and every face contains a subset of the extreme points of P by the extremeexistence theorem (2.6.0.0.2). This means: because the affine subset A and hyperplane{X | 〈C , X 〉 = f ⋆ 0 } must intersect a whole face of P , calculation of an upper bound onrank of X ⋆ ignores counting the hyperplane when determining m in (272).