v2010.10.26 - Convex Optimization

370 CHAPTER 4. SEMIDEFINITE PROGRAMMINGis equivalent to cardinality c of vector x . Then we can transform cardinalityproblem (829) to an equivalent problem in new variable X : 4.44maximizeX∈S N 〈X , A〉subject to 〈X , I〉 = 1(X ≽ 0)rankX = 1card δ(X) ≤ c(832)We transform problem (832) to an equivalent convex problem byintroducing two direction matrices into regularization terms: W to achievedesired cardinality cardδ(X) , and 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(833)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 semidefiniteprogramminimize 〈X ⋆ , Y 〉Y ∈ S Nsubject to 0 ≼ Y ≼ I(834)trY = N − 1and 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(835)Both direction matrix programs are derived from (1700a) whose analyticalsolution is known but is not necessarily unique. We emphasize (confer p.309):because this iteration (833) (834) (835) (initial Y,W = 0) is not a projection4.44 A semidefiniteness constraint X ≽ 0 is not required, theoretically, because positivesemidefiniteness of a rank-1 matrix is enforced by symmetry. (Theorem A.3.1.0.7)