v2007.09.17 - Convex Optimization

7.3. THIRD PREVALENT PROBLEM: 477Our plan is to instead divide problem (1211) into two and then alternatetheir solution:∥minimize ∥δ(Υ) − π ( δ(R T ΛR) )∥ ∥ 2Υ ⎡ ⎤subject to δ(Υ) ∈ ⎣ Rρ+1 +(a)0 ⎦ ∩ ∂HR −(1213)minimize ‖RΥR T − Λ‖ 2 FRsubject to δ(QRΥR T Q T ) = 0R −1 = R T(b)We justify disappearance of the hollowness constraint in convexoptimization problem (1213a): From the arguments in7.1.3 withregard ⎡to π ⎤the permutation operator, cone membership constraintδ(Υ) ∈⎣ Rρ+1 +0 ⎦∩ ∂H from (1213a) is equivalent toR −⎡ ⎤δ(Υ) ∈ ⎣ Rρ+1 +0 ⎦ ∩ ∂H ∩ K M (1214)R −where K M is the monotone cone (2.13.9.4.2). Membership of δ(Υ) to thepolyhedral cone of majorization (Theorem A.1.2.0.1)K ∗ λδ = ∂H ∩ K ∗ M+ (1229)where K ∗ M+ is the dual monotone nonnegative cone (2.13.9.4.1), is acondition (in absence of a hollowness [ constraint) ] that would insure existence0 1Tof a symmetric hollow matrix . Curiously, intersection of1 −D⎡ ⎤this feasible superset ⎣ Rρ+1 +0 ⎦∩ ∂H ∩ K M from (1214) with the cone ofmajorization K ∗ λδR −is a benign operation; id est,∂H ∩ K ∗ M+ ∩ K M = ∂H ∩ K M (1215)