v2010.10.26 - Convex Optimization

7.2. SECOND PREVALENT PROBLEM: 573where κ ∈ R + is a constant determined by cut-and-try. The equivalentsemidefinite program makes κ variable: for nonnegative and symmetric HminimizeD , Y , κsubject toκρ + 2 tr(V Y V )[ ]dij y ij≽ 0 ,y ijh 2 ijN ≥ j > i = 1... N −1(1370)− tr(V DV ) ≤ κρY ∈ S N hD ∈ EDM Nwhich is the same as (1359), the problem with no explicit constraint on affinedimension. As the present problem is stated, the desired affine dimension ρyields to the variable scale factor κ ; ρ is effectively ignored.Yet this result is an illuminant for problem (1359) and it equivalents(all the way back to (1352)): When the given measurement matrix His nonnegative and symmetric, finding the closest EDM D as in problem(1352), (1355), or (1359) implicitly entails minimization of affine dimension(confer5.8.4,5.14.4). Those non−rank-constrained problems are eachinherently equivalent to cenv(rank)-minimization problem (1370), in otherwords, and their optimal solutions are unique because of the strictly convexobjective function in (1352).7.2.2.3 Rank-heuristic insightMinimization of affine dimension by use of this trace rank-heuristic (1368)tends to find a list configuration of least energy; rather, it tends to optimizecompaction of the reconstruction by minimizing total distance. (915) It is bestused where some physical equilibrium implies such an energy minimization;e.g., [352,5].For this Problem 2, the trace rank-heuristic arose naturally in theobjective in terms of V . We observe: V (in contrast to VN T ) spreads energyover all available distances (B.4.2 no.20, contrast no.22) although the rankfunction itself is insensitive to choice of auxiliary matrix.Trace rank-heuristic (1364) is useless when a main diagonal is constrainedto be constant. Such would be the case were optimization over anelliptope (5.4.2.3), or when the diagonal represents a Boolean vector; e.g.,Example 4.2.3.1.1, Example 4.6.0.0.8.