v2006.03.09 - Convex Optimization

7.3. THIRD PREVALENT PROBLEM: 4157.3.1.1.1 Example. Alternating projection on nearest EDM.By solving (1003) we confirm the result from an example given by Glunt,Hayden, et alii [85,6] who found an analytical solution to the convexoptimization problem (1000) for the particular cardinality N = 3 by usingthe alternating projection method of von Neumann (E.10):⎡H = ⎣0 1 11 0 91 9 0⎤⎦ , D ⋆ =⎡⎢⎣19 1909 919 7609 91997690⎤⎥⎦ (1005)The original problem (1000) of projecting H on the EDM cone is transformedto an equivalent iterative sequence of projections on the two convex cones(792) from5.8.1.1. 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 (1003) can be reformulated by moving the objectivefunction inminimize ‖D − H‖ 2 FD(1000)subject to D ∈ EDM Nto the constraints. This makes an equivalent second-order cone program: forany measurement matrix Hminimize tt∈R , Dsubject to ‖D − H‖ 2 F ≤ t(1006)D ∈ EDM NWe can transform this problem to an equivalent Schur-form semidefiniteprogram; (A.4.1)minimizet∈R , Dsubject tot[]tI vec(D − H)vec(D − H) T ≽ 0 (1007)1D ∈ EDM N