v2007.09.13 - Convex Optimization

458 CHAPTER 7. PROXIMITY PROBLEMS7.2.1.1 Equivalent semidefinite program, Problem 2, convex caseConvex problem (1154) is numerically solvable for its global minimum usingan interior-point method [46,11] [296] [213] [203] [287] [9] [97]. We translate(1154) to an equivalent semidefinite program (SDP) for a pedagogical reasonmade clear in7.2.2.2 and because there exist readily available computerprograms for numerical solution [251] [116] [267] [27] [288] [289] [290].Substituting a new matrix variable Y ∆ = [y ij ]∈ R N×N+h ij√dij ← y ij (1156)Boyd proposes: problem (1154) is equivalent to the semidefinite program∑minimize d ij − 2y ij + h 2 ijD , Yi,j[ ]dij y ij(1157)subject to≽ 0 , i,j =1... Ny ij h 2 ijD ∈ EDM NTo see that, recall d ij ≥ 0 is implicit to D ∈ EDM N (5.8.1, (724)). Sowhen H ∈ R N×N+ is nonnegative as assumed,[ ]dij y √ij≽ 0 ⇔ h ij√d ij ≥ yij 2 (1158)y ijh 2 ijMinimization of the objective function implies maximization of y ij that isbounded above. √Hence nonnegativity of y ij is implicit to (1157) and, asdesired, y ij →h ij dij as optimization proceeds.If the given matrix H is now assumed symmetric and nonnegative,H = [h ij ] ∈ S N ∩ R N×N+ (1159)then Y = H ◦ ◦√ D must belong to K= S N h ∩ R N×N+ (1103). Because Y ∈ S N h(B.4.2 no.20), then‖ ◦√ D − H‖ 2 F = ∑ i,jd ij − 2y ij + h 2 ij = −N tr(V (D − 2Y )V ) + ‖H‖ 2 F (1160)