v2007.09.17 - Convex Optimization

7.3. THIRD PREVALENT PROBLEM: 471a matrix of distance-square squared,minimize − tr(V (∂ − 2H ◦D)V )∂ , D[ ]∂ij d ijsubject to≽ 0 , j > i = 1... N −1d ij 1(1193)D ∈ EDM N∂ ∈ S N hwhere [∂ij d ijd ij 1]≽ 0 ⇔ ∂ ij ≥ d 2 ij (1194)Symmetry of input H facilitates trace in the objective (B.4.2 no.20), whileits nonnegativity causes ∂ ij →d 2 ij as optimization proceeds.7.3.1.1.1 Example. Alternating projection on nearest EDM.By solving (1193) we confirm the result from an example given by Glunt,Hayden, et alii [106,6] who found an analytical solution to convexoptimization problem (1189) for particular cardinality N = 3 by using thealternating projection method of von Neumann (E.10):⎡H = ⎣0 1 11 0 91 9 0⎤⎦ , D ⋆ =⎡⎢⎣19 1909 919 7609 919 7609 9⎤⎥⎦ (1195)The original problem (1189) of projecting H on the EDM cone is transformedto an equivalent iterative sequence of projections on the two convex cones(1056) from6.8.1.1. Using ordinary alternating projection, input H goes toD ⋆ with an accuracy of four decimal places in about 17 iterations. Affinedimension corresponding to this optimal solution is r = 1.Obviation of semidefinite programming’s computational expense is theprincipal advantage of this alternating projection technique. 7.3.1.2 Schur-form semidefinite program, Problem 3 convex caseSemidefinite program (1193) can be reformulated by moving the objectivefunction inminimize ‖D − H‖ 2 FD(1189)subject to D ∈ EDM N