v2007.09.13 - Convex Optimization
v2007.09.13 - Convex Optimization v2007.09.13 - Convex Optimization
468 CHAPTER 7. PROXIMITY PROBLEMScone and because the objective function‖D − H‖ 2 F = ∑ i,j(d ij − h ij ) 2 (1186)is a strictly convex quadratic in D ; 7.15∑minimize d 2 ij − 2h ij d ij + h 2 ijDi,j(1187)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) [105] [98] [133,1] We translate(1187) to an equivalent semidefinite program because we have a good solver:Assume the given measurement matrix H to be nonnegative andsymmetric; 7.16H = [h ij ] ∈ S N ∩ R N×N+ (1159)We then propose: Problem (1187) is equivalent to the semidefinite program,for∂ = ∆ [d 2 ij ] = D ◦D (1188)7.15 For nonzero Y ∈ S N h and some open interval of t∈R (3.2.3.0.2,D.2.3)d 2dt 2 ‖(D + tY ) − H‖2 F = 2 trY T Y > 07.16 If that H given has negative entries, then the technique of solution presented herebecomes invalid. Projection of H on K (1103) prior to application of this proposedtechnique, as explained in7.0.1, is incorrect.
7.3. THIRD PREVALENT PROBLEM: 469a matrix of distance-square squared,minimize − tr(V (∂ − 2H ◦D)V )∂ , D[ ]∂ij d ijsubject to≽ 0 , j > i = 1... N −1d ij 1(1189)D ∈ EDM N∂ ∈ S N hwhere [∂ij d ijd ij 1]≽ 0 ⇔ ∂ ij ≥ d 2 ij (1190)Symmetry of input H facilitates trace in the objective (B.4.2 no.20), whileits nonnegativity causes ∂ ij →d 2 ij as optimization proceeds.7.3.1.1.1 Example. Alternating projection on nearest EDM.By solving (1189) we confirm the result from an example given by Glunt,Hayden, et alii [105,6] who found an analytical solution to convexoptimization problem (1185) for particular cardinality N = 3 by using thealternating projection method of von Neumann (E.10):⎡H = ⎣0 1 11 0 91 9 0⎤⎦ , D ⋆ =⎡⎢⎣19 1909 919 7609 919 7609 9⎤⎥⎦ (1191)The original problem (1185) of projecting H on the EDM cone is transformedto an equivalent iterative sequence of projections on the two convex cones(1052) from6.8.1.1. Using ordinary alternating projection, input H goes toD ⋆ with an accuracy of four decimal places in about 17 iterations. Affinedimension corresponding to this optimal solution is r = 1.Obviation of semidefinite programming’s computational expense is theprincipal advantage of this alternating projection technique. 7.3.1.2 Schur-form semidefinite program, Problem 3 convex caseSemidefinite program (1189) can be reformulated by moving the objectivefunction inminimize ‖D − H‖ 2 FD(1185)subject to D ∈ EDM N
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- Page 423 and 424: 6.8. DUAL EDM CONE 423EDM 2 = S 2 h
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468 CHAPTER 7. PROXIMITY PROBLEMScone and because the objective function‖D − H‖ 2 F = ∑ i,j(d ij − h ij ) 2 (1186)is a strictly convex quadratic in D ; 7.15∑minimize d 2 ij − 2h ij d ij + h 2 ijDi,j(1187)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) [105] [98] [133,1] We translate(1187) to an equivalent semidefinite program because we have a good solver:Assume the given measurement matrix H to be nonnegative andsymmetric; 7.16H = [h ij ] ∈ S N ∩ R N×N+ (1159)We then propose: Problem (1187) is equivalent to the semidefinite program,for∂ = ∆ [d 2 ij ] = D ◦D (1188)7.15 For nonzero Y ∈ S N h and some open interval of t∈R (3.2.3.0.2,D.2.3)d 2dt 2 ‖(D + tY ) − H‖2 F = 2 trY T Y > 07.16 If that H given has negative entries, then the technique of solution presented herebecomes invalid. Projection of H on K (1103) prior to application of this proposedtechnique, as explained in7.0.1, is incorrect.