v2007.09.13 - Convex Optimization

466 CHAPTER 7. PROXIMITY PROBLEMSthat becomes equivalent to (1178) by making explicit the constraint on affinedimension via rank. The iteration is formed by moving the dimensionalconstraint to the objective:minimize −〈V (D − 2Y )V , I〉 − w〈VN TDV N , W 〉D , Y[ ]dij y ijsubject to≽ 0 , j > i = 1... N −1y ij h 2 ij(1180)Y ∈ S N hD ∈ EDM Nwhere w (≈ 10) is a positive scalar just large enough to make 〈V T N DV N , W 〉vanish to within some numerical precision, and where direction matrix W isan optimal solution to semidefinite program (1475a)minimize −〈VN T WD⋆ V N , W 〉subject to 0 ≼ W ≼ ItrW = N − 1(1181)known in closed form. Semidefinite programs (1180) and (1181) are iterateduntil convergence in the sense defined on page 257. This iteration is nota projection method. Convex problem (1180) is neither a relaxation ofunidimensional scaling problem (1179); instead, problem (1180) is a convexequivalent to (1179) at convergence of the iteration.Jan de Leeuw provided us with some test data⎡⎤0.000000 5.235301 5.499274 6.404294 6.486829 6.2632655.235301 0.000000 3.208028 5.840931 3.559010 5.353489H =5.499274 3.208028 0.000000 5.679550 4.020339 5.239842⎢ 6.404294 5.840931 5.679550 0.000000 4.862884 4.543120⎥⎣ 6.486829 3.559010 4.020339 4.862884 0.000000 4.618718 ⎦6.263265 5.353489 5.239842 4.543120 4.618718 0.000000(1182)and a globally optimal solutionX ⋆ = [ −4.981494 −2.121026 −1.038738 4.555130 0.764096 2.822032 ]= [ x ⋆ 1 x ⋆ 2 x ⋆ 3 x ⋆ 4 x ⋆ 5 x ⋆ 6 ](1183)