v2007.11.26 - Convex Optimization

228 CHAPTER 4. SEMIDEFINITE PROGRAMMINGwhere svec X defined by (47) denotes symmetric vectorization.Semidefinite programming has emerged recently to prominence primarilybecause it admits a new class of problem previously unsolvable by convexoptimization techniques, [44] secondarily because it theoretically subsumesother convex techniques such as linear, quadratic, and second-order coneprogramming. Determination of the Riemann mapping function fromcomplex analysis [213] [24,8,13], for example, can be posed as asemidefinite program.4.1.1 Maximal complementarityIt has been shown that contemporary interior-point methods (developedcirca 1990 [102]) [46,11] [292] [216] [206] [9] [98] for numericalsolution of semidefinite programs can converge to a solution of maximalcomplementarity; [126,5] [300] [186] [109] not a vertex-solution but asolution of highest cardinality or rank among all optimal solutions. 4.2[301,2.5.3]4.1.1.1 Reduced-rank solutionA simple rank reduction algorithm, for construction of a primal optimalsolution X ⋆ to (547P) satisfying an upper bound on rank governed byProposition 2.9.3.0.1, is presented in4.3. That proposition asserts existenceof feasible solutions with an upper bound on their rank; [20,II.13.1]specifically, it asserts an extreme point (2.6.0.0.1) of the primal feasibleset A ∩ S n + satisfies upper bound⌊√ ⌋ 8m + 1 − 1rankX ≤2where, given A∈ R m×n(n+1)/2 and b∈ R m(233)A ∆ = {X ∈ S n | A svec X = b} (550)is the affine subset from primal problem (547P).4.2 This characteristic might be regarded as a disadvantage to this method of numericalsolution, but this behavior is not certain and depends on solver implementation.