v2006.03.09 - Convex Optimization

414 CHAPTER 7. EDM PROXIMITY∑minimize d 2 ij − 2h ij d ij + h 2 ijDi,j(1002)subject to D ∈ EDM NOptimal solution D ⋆ is therefore unique, as expected, for this simpleprojection on the EDM cone.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) [85] [79] [110,1] We translate(1002) to an equivalent semidefinite program because we have a good solver:Assume the given measurement matrix H to be nonnegative andsymmetric; 7.17H = [h ij ] ∈ S N ∩ R N×N+ (983)We then propose: Problem (1002) is equivalent to the semidefinite program,for ∂ = ∆ [d 2 ij ]=D ◦D distance-square squared,minimize − tr(V (∂ − 2H ◦D)V )∂ , D[ ]∂ij d ijsubject to≽ 0 , j > i = 1... N −1d ij 1(1003)whereD ∈ EDM N∂ ∈ S N h[∂ij d ijd ij 1]≽ 0 ⇔ ∂ ij ≥ d 2 ij (1004)Symmetry of input H facilitates trace in the objective (B.4.2 no.20), whileits nonnegativity causes ∂ ij →d 2 ij as optimization proceeds.7.17 If that H given has negative entries, then the technique of solution presented herebecomes invalid. Projection of H on K (929) prior to application of this proposedtechnique, as explained in7.0.1, is incorrect.