v2007.09.13 - Convex Optimization

7.1. FIRST PREVALENT PROBLEM: 4477.1.1 Closest-EDM Problem 1, convex case7.1.1.0.1 Proof. Solution (1118), convex case.When desired affine dimension is unconstrained, ρ =N −1, the rank functiondisappears from (1116) leaving a convex optimization problem; a simpleunique minimum-distance projection on the positive semidefinite cone S N−1videlicetby (724). Becauseminimize ‖−VD∈ S N N T(D − H)V N ‖ 2 Fhsubject to −VN TDV N ≽ 0S N−1 = −V T N S N h V N (816)+ :(1121)then the necessary and sufficient conditions for projection in isometricallyisomorphic R N(N−1)/2 on the self-dual (321) positive semidefinite cone S N−1+are: 7.6 (E.9.2.0.1) (1370) (confer (1795))−V T N D⋆ V N ≽ 0−V T N D⋆ V N(−VTN D ⋆ V N + V T N HV N)= 0−V T N D⋆ V N + V T N HV N ≽ 0(1122)Symmetric −VN THV N is diagonalizable hence decomposable in terms of itseigenvectors v and eigenvalues λ as in (1119). Therefore (confer (1118))−V T ND ⋆ V N =N−1∑i=1max{0, λ i }v i v T i (1123)satisfies (1122), optimally solving (1121). To see that, recall: theseeigenvectors constitute an orthogonal set and−V T ND ⋆ V N + V T NHV NN−1∑= − min{0, λ i }v i vi T (1124)i=17.6 These conditions for projection on a convex cone are identical to theKarush-Kuhn-Tucker (KKT) optimality conditions for problem (1121).