v2010.10.26 - Convex Optimization

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 311S n(I −W)G ⋆ (I −W)G ⋆WG ⋆ WS ⊥ nFigure 83: (confer Figure 165) Projection of G ⋆ on subspace S n of rank≤nmatrices whose nullspace contains N(G ⋆ ). This direction W is closed-formsolution to (1700a).When direction matrix W = I , as in the trace heuristic for example,then −W points directly at the origin (the rank-0 PSD matrix, Figure 84).Vector inner-product of an optimization variable with direction matrix Wis therefore a generalization of the trace heuristic (7.2.2.1) for rankminimization; −W is instead trained toward the boundary of the positivesemidefinite cone.4.4.1.2 convergenceWe study convergence to ascertain conditions under which a direction matrixwill reveal a feasible solution G , of rank n or less, to semidefinite program(736). Denote by W ⋆ a particular optimal direction matrix from semidefiniteprogram (1700a) such that (737) holds (feasible rankG≤n found). Then wedefine global convergence of the iteration (736) (1700a) to correspond withthis vanishing vector inner-product (737) of optimal solutions.Because this iterative technique for constraining rank is not a projectionmethod, it can find a rank-n solution G ⋆ ((737) will be satisfied) only if atleast one exists in the feasible set of program (736).