v2007.09.13 - Convex Optimization

286 CHAPTER 4. SEMIDEFINITE PROGRAMMINGcardδ(X) , and direction matrix Y to find an approximating rank-onematrix X :maximize 〈X , A − w 1 Y 〉 − w 2 〈δ(X) , δ(W)〉X∈S Nsubject to 〈X , I〉 = 1X ≽ 0(690)where w 1 and w 2 are positive scalars respectively weighting tr(XY ) andδ(X) T δ(W) just enough to insure that they vanish to within some numericalprecision, where direction matrix Y is an optimal solution to semidefiniteprogramminimizeY ∈ S N 〈X ⋆ , Y 〉subject to 0 ≼ Y ≼ ItrY = N − 1(691)and where diagonal direction matrix W ∈ S N optimally solves linear programminimize 〈δ(X ⋆ ) , δ(W)〉W=δ 2 (W)subject to 0 ≼ δ(W) ≼ 1trW = N − c(692)both direction matrix programs being derived from (1475a) whose analyticalsolution is known. We emphasize (confer p.257): because this iteration(690) (691) (692) (initial Y,W = 0) is not a projection method, success relieson existence of matrices in the feasible set of (690) having desired rank anddiagonal cardinality. In particular, the feasible set of convex problem (690)is a Fantope (80) whose extreme points constitute the set of all normalizedrank-one matrices; among those are found rank-one matrices of any desireddiagonal cardinality.Convex problem (690) is neither a relaxation of cardinality problem (686);instead, problem (690) is a convex equivalent to (686) at convergence ofiteration (690) (691) (692). Because the feasible set of convex problem (690)contains all normalized rank-one matrices of any desired diagonal cardinality,a constraint too low or high in cardinality will not prevent solution. Anoptimal solution, whose diagonal cardinality is equal to cardinality of aprincipal eigenvector of matrix A , will produce the lowest residual Frobeniusnorm (to within machine precision and other noise processes) in the originalproblem statement (685).