v2007.09.13 - Convex Optimization

4.1. CONIC PROBLEM 2334.1.1.3 Previous workBarvinok showed [21,2.2] when given a positive definite matrix C and anarbitrarily small neighborhood of C comprising positive definite matrices,there exists a matrix ˜C from that neighborhood such that optimal solutionX ⋆ to (546P) (substituting ˜C) is an extreme point of A ∩ S n + and satisfiesupper bound (232). 4.4 Given arbitrary positive definite C , this meansnothing inherently guarantees that an optimal solution X ⋆ to problem (546P)satisfies (232); certainly nothing given any symmetric matrix C , as theproblem is posed. This can be proved by example:4.1.1.3.1 Example. (Ye) Maximal Complementarity.Assume dimension n to be an even positive number. Then the particularinstance of problem (546P),has optimal solutionminimizeX∈ S n 〈[ I 00 2Isubject to X ≽ 0X ⋆ =〈I , X〉 = n[ 2I 00 0] 〉, X(556)]∈ S n (557)with an equal number of twos and zeros along the main diagonal. Indeed,optimal solution (557) is a terminal solution along the central path taken bythe interior-point method as implemented in [296,2.5.3]; it is also a solutionof highest rank among all optimal solutions to (556). Clearly, rank of thisprimal optimal solution exceeds by far a rank-1 solution predicted by upperbound (232).4.1.1.4 Later developmentsThis rational example (556) indicates the need for a more generally applicableand simple algorithm to identify an optimal solution X ⋆ satisfying Barvinok’sProposition 2.9.3.0.1. We will review such an algorithm in4.3, but first weprovide more background.4.4 Further, the set of all such ˜C in that neighborhood is open and dense.