v2010.10.26 - Convex Optimization

7.1. FIRST PREVALENT PROBLEM: 5577.1.1 Closest-EDM Problem 1, convex case7.1.1.0.1 Proof. Solution (1316), convex case.When desired affine dimension is unconstrained, ρ =N −1, the rank functiondisappears from (1314) leaving a convex optimization problem; a simpleunique minimum-distance projection on the positive semidefinite cone S N−1videlicetby (910). 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 (1011)+ :(1319)then the necessary and sufficient conditions for projection in isometricallyisomorphic R N(N−1)/2 on the selfdual (377) positive semidefinite cone S N−1+are: 7.8 (E.9.2.0.1) (1590) (confer (2031))−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(1320)Symmetric −VN THV N is diagonalizable hence decomposable in terms of itseigenvectors v and eigenvalues λ as in (1317). Therefore (confer (1316))−V T ND ⋆ V N =N−1∑i=1max{0, λ i }v i v T i (1321)satisfies (1320), optimally solving (1319). 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 (1322)i=17.8 The Karush-Kuhn-Tucker (KKT) optimality conditions [280, p.328] [61,5.5.3] forproblem (1319) are identical to these conditions for projection on a convex cone.