v2004.06.19 - Convex Optimization

228 CHAPTER 6. CONVEX OPTIMIZATIONThe vector inner product for matrices is defined in theEuclidean/Frobenius sense in the isomorphic vector space R n2 ; id est,〈C,X〉 =tr(C ∆ T X)=vec(C) T vec X where vec X (18) denotes vectorizationby stacking columns in the natural order.It has been shown that interior-point methods [1, §11] [42] [121] [6] canconverge to a solution of maximal complementarity; [122] (§6.1.2.3.1) inthat case, not a vertex-solution but a solution of high cardinality or rank.[4, §2.5.3]We desire a simple algorithm for construction of a primal optimal solutionX ⋆ to (P) satisfying a least upper bound on rank governed byProposition 2.6.6.4.1 (Barvinok) that asserts existence of low-rank feasiblesolutions. [23, §II.13.1] Specifically, the proposition asserts an extreme point(§2.5) of the primal feasible set A ∩ S n + satisfies least upper bound⌊√ ⌋ 8m + 1 − 1rankX ≤(148)2whereA ∆ = {X ∈ S n | A vecX = b} (648)is the affine set from primal problem (P). Barvinok showed [60, §2.2] whengiven a positive definite matrix C and an arbitrarily small neighborhood ofC comprising positive definite matrices, there exists a matrix ˜C from thatneighborhood such that optimal solution X ⋆ to (P) (substituting ˜C) is anextreme point of A ∩ S n + and satisfies least upper bound (148). 6.4This means given arbitrary positive definite C , there is no guarantee anoptimal solution X ⋆ to (P) (using C) satisfies (148).To prove that by example: (Ye) Assume dimension n to be an evenpositive number. Then the particular instance of problem (P),〈[ ] 〉 I 0minimize , XX 0 2I(649)subject to X ≽ 0has optimal solutionX ⋆ =〈I , X〉 = n[ 2I 00 0]∈ S n (650)6.4 Further, the set of all such ˜C in that neighborhood is open and dense.