v2007.09.17 - Convex Optimization

462 CHAPTER 7. PROXIMITY PROBLEMSConfinement of G to the geometric center subspace provides numericalstability and no loss of generality (confer (993)); implicit constraint G1 = 0is otherwise unnecessary.To include constraints on the list X ∈ R n×N , we would first rewrite (1166)minimize − tr(V (D(G) − 2Y )V )G∈S N c , Y ∈ S N h , X∈ Rn×N [ ]〈Φij , G〉 y ijsubject to≽ 0 ,y ijh 2 ij[ ] I XX T ≽ 0GX ∈ Cj > i = 1... N −1(1167)and then add the constraints, realized here in abstract membership to someconvex set C . This problem realization includes a convex relaxation of thenonconvex constraint G = X T X and, if desired, more constraints on G couldbe added. This technique is discussed in5.4.2.2.4.7.2.2 Minimization of affine dimension in Problem 2When desired affine dimension ρ is diminished, the rank function becomesreinserted into problem (1161) that is then rendered difficult to solve becausethe feasible set {D , Y } loses convexity in S N h × R N×N . Indeed, the rankfunction is quasiconcave (3.3) on the positive semidefinite cone; (2.9.2.6.2)id est, its sublevel sets are not convex.7.2.2.1 Rank minimization heuristicA remedy developed in [91] [192] [92] [90] introduces convex envelope (cenv)of the quasiconcave rank function: (Figure 111)7.2.2.1.1 Definition. Convex envelope. [147]The convex envelope of a function f : C →R is defined as the largest convexfunction g such that g ≤ f on convex domain C ⊆ R n . 7.14 △7.14 Provided f ≢+∞ and there exists an affine function h ≤f on R n , then the convexenvelope is equal to the convex conjugate (the Legendre-Fenchel transform) of the convexconjugate of f ; id est, the conjugate-conjugate function f ∗∗ . [148,E.1]