v2010.10.26 - Convex Optimization

586 CHAPTER 7. PROXIMITY PROBLEMSProof. We justify disappearance of the hollowness constraint inconvex optimization problem (1408a): From the arguments in7.1.3with regard ⎡ to⎤π the permutation operator, cone membership constraintδ(Υ) ∈⎣ Rρ+1 +0 ⎦∩ ∂H from (1408a) is equivalent toR −⎡δ(Υ) ∈ ⎣ Rρ+1 +0R −⎤⎦ ∩ ∂H ∩ K M (1409)where K M is the monotone cone (2.13.9.4.3). Membership of δ(Υ) to thepolyhedral cone of majorization (Theorem A.1.2.0.1)K ∗ λδ = ∂H ∩ K ∗ M+ (1426)where K ∗ M+ is the dual monotone nonnegative cone (2.13.9.4.2), 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 (1409) with the cone ofmajorization K ∗ λδR −is a benign operation; id est,∂H ∩ K ∗ M+ ∩ K M = ∂H ∩ K M (1410)verifiable by observing conic dependencies (2.10.3) among the aggregate ofhalfspace-description normals. The cone membership constraint in (1408a)therefore inherently insures existence of a symmetric hollow matrix.