v2010.10.26 - Convex Optimization

278 CHAPTER 4. SEMIDEFINITE PROGRAMMINGNevertheless, semidefinite programming has recently emerged toprominence because it admits a new class of problem previously unsolvable byconvex optimization techniques, [59] and because it theoretically subsumesother convex techniques: (Figure 80) linear programming and quadraticprogramming and second-order cone programming. 4.4 Determination of theRiemann mapping function from complex analysis [286] [29,8,13], forexample, can be posed as a semidefinite program.4.1.2 Maximal complementarityIt has been shown [391,2.5.3] that contemporary interior-point methods[382] [289] [277] [11] [61,11] [145] (developed circa 1990 [151] for numericalsolution of semidefinite programs) can converge to a solution of maximalcomplementarity; [176,5] [390] [253] [158] not a vertex-solution but asolution of highest cardinality or rank among all optimal solutions. 4.5This phenomenon can be explained by recognizing that interior-pointmethods generally find solutions relatively interior to a feasible set bydesign. 4.6 [6, p.3] Log barriers are designed to fail numerically at the feasibleset boundary. So low-rank solutions, all on the boundary, are rendered moredifficult to find as numerical error becomes more prevalent there.4.1.2.1 Reduced-rank solutionA simple rank reduction algorithm, for construction of a primal optimalsolution X ⋆ to (649P) 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; [26,II.13.1]specifically, it asserts an extreme point (2.6.0.0.1) of primal feasible setA ∩ S n + satisfies upper bound⌊√ ⌋ 8m + 1 − 1rankX ≤(272)2where, given A∈ R m×n(n+1)/2 and b∈ R m ,A {X ∈ S n | A svec X = b} (652)4.4 SOCP came into being in the 1990s; it is not posable as a quadratic program. [248]4.5 This characteristic might be regarded as a disadvantage to this method of numericalsolution, but this behavior is not certain and depends on solver implementation.4.6 Simplex methods, in contrast, generally find vertex solutions.