v2010.10.26 - Convex Optimization

4.1. CONIC PROBLEM 2834.1.2.3 Previous workBarvinok showed, [24,2.2] 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 (649P) (substituting ˜C) is an extreme point of A ∩ S n + and satisfiesupper bound (272). 4.8 Given arbitrary positive definite C , this meansnothing inherently guarantees that an optimal solution X ⋆ to problem (649P)satisfies (272); certainly nothing given any symmetric matrix C , as theproblem is posed. This can be proved by example:4.1.2.3.1 Example. (Ye) Maximal Complementarity.Assume dimension n to be an even positive number. Then the particularinstance of problem (649P),has optimal solution〈[ I 0minimizeX∈ S n 0 2Isubject to X ≽ 0X ⋆ =〈I , X〉 = n[ 2I 00 0] 〉, X(659)]∈ S n (660)with an equal number of twos and zeros along the main diagonal. Indeed,optimal solution (660) is a terminal solution along the central path taken bythe interior-point method as implemented in [391,2.5.3]; it is also a solutionof highest rank among all optimal solutions to (659). Clearly, rank of thisprimal optimal solution exceeds by far a rank-1 solution predicted by upperbound (272).4.1.2.4 Later developmentsThis rational example (659) 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.4.8 Further, the set of all such ˜C in that neighborhood is open and dense.