v2010.10.26 - Convex Optimization

7.3. THIRD PREVALENT PROBLEM: 579is a strictly convex quadratic in D ; 7.18∑minimize d 2 ij − 2h ij d ij + h 2 ijDi,j(1386)subject to D ∈ EDM NOptimal solution D ⋆ is therefore unique, as expected, for this simpleprojection on the EDM cone equivalent to (1309).7.3.1.1 Equivalent semidefinite program, Problem 3, convex caseIn the past, this convex problem was solved numerically by means ofalternating projection. (Example 7.3.1.1.1) [155] [148] [185,1] We translate(1386) to an equivalent semidefinite program because we have a good solver:Assume the given measurement matrix H to be nonnegative andsymmetric; 7.19H = [h ij ] ∈ S N ∩ R N×N+ (1357)We then propose: Problem (1386) is equivalent to the semidefinite program,for∂ [d 2 ij ] = D ◦D (1387)a matrix of distance-square squared,minimize − tr(V (∂ − 2H ◦D)V )∂ , D[ ]∂ij d ijsubject to≽ 0 , N ≥ j > i = 1... N −1d ij 1(1388)D ∈ EDM N∂ ∈ S N h7.18 For nonzero Y ∈ S N h and some open interval of t∈R (3.7.3.0.2,D.2.3)d 2dt 2 ‖(D + tY ) − H‖2 F = 2 tr Y T Y > 07.19 If that H given has negative entries, then the technique of solution presented herebecomes invalid. Projection of H on K (1301) prior to application of this proposedtechnique, as explained in7.0.1, is incorrect.