v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
566 CHAPTER 7. PROXIMITY PROBLEMSby (1704) where{ ∣µ ⋆ = maxλ ( −VN T (D ⋆ − H)V N∣ , i = 1... N −1i)i} ∈ R + (1346)the minimized largest absolute eigenvalue (due to matrix symmetry).For lack of a unique solution here, we prefer the Frobenius rather thanspectral norm.7.2 Second prevalent problem:Projection on EDM cone in √ d ijLet◦√D [√dij ] ∈ K = S N h ∩ R N×N+ (1347)be an unknown matrix of absolute distance; id est,D = [d ij ] ◦√ D ◦ ◦√ D ∈ EDM N (1348)where ◦ denotes Hadamard product. The second prevalent proximityproblem is a Euclidean projection (in the natural coordinates √ d ij ) of matrixH on a nonconvex subset of the boundary of the nonconvex cone of Euclideanabsolute-distance matrices rel∂ √ EDM N : (6.3, confer Figure 138b)minimize ‖ ◦√ ⎫D − H‖◦√ 2 FD⎪⎬subject to rankVN TDV N ≤ ρ Problem 2 (1349)◦√ √D ∈ EDMN⎪⎭where√EDM N = { ◦√ D | D ∈ EDM N } (1182)This statement of the second proximity problem is considered difficult tosolve because of the constraint on desired affine dimension ρ (5.7.2) andbecause the objective function‖ ◦√ D − H‖ 2 F = ∑ i,j( √ d ij − h ij ) 2 (1350)is expressed in the natural coordinates; projection on a doubly nonconvexset.
7.2. SECOND PREVALENT PROBLEM: 567Our solution to this second problem prevalent in the literature requiresmeasurement matrix H to be nonnegative;H = [h ij ] ∈ R N×N+ (1351)If the H matrix given has negative entries, then the technique of solutionpresented here becomes invalid. As explained in7.0.1, projection of Hon K = S N h ∩ R N×N+ (1301) prior to application of this proposed solution isincorrect.7.2.1 Convex caseWhen ρ = N − 1, the rank constraint vanishes and a convex problem thatis equivalent to (1308) emerges: 7.14minimize◦√Dsubject to‖ ◦√ D − H‖ 2 F◦√D ∈√EDMN⇔minimizeD∑ √d ij − 2h ij dij + h 2 iji,jsubject to D ∈ EDM N (1352)For any fixed i and j , the argument of summation is a convex functionof d ij because (for nonnegative constant h ij ) the negative square root isconvex in nonnegative d ij and because d ij + h 2 ij is affine (convex). Becausethe sum of any number of convex functions in D remains convex [61,3.2.1]and because the feasible set is convex in D , we have a convex optimizationproblem:minimize 1 T (D − 2H ◦ ◦√ D )1 + ‖H‖ 2 FD(1353)subject to D ∈ EDM NThe objective function being a sum of strictly convex functions is,moreover, strictly convex in D on the nonnegative orthant. Existenceof a unique solution D ⋆ for this second prevalent problem depends uponnonnegativity of H and a convex feasible set (3.1.2). 7.157.14 still thought to be a nonconvex problem as late as 1997 [354] even though discoveredconvex by de Leeuw in 1993. [103] [53,13.6] Yet using methods from3, it can be easilyascertained: ‖ ◦√ D − H‖ F is not convex in D .7.15 The transformed problem in variable D no longer describes √ Euclidean projection onan EDM cone. Otherwise we might erroneously conclude EDM N were a convex bodyby the Bunt-Motzkin theorem (E.9.0.0.1).
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- Page 533 and 534: 6.8. DUAL EDM CONE 533EDM 2 = S 2 h
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566 CHAPTER 7. PROXIMITY PROBLEMSby (1704) where{ ∣µ ⋆ = maxλ ( −VN T (D ⋆ − H)V N∣ , i = 1... N −1i)i} ∈ R + (1346)the minimized largest absolute eigenvalue (due to matrix symmetry).For lack of a unique solution here, we prefer the Frobenius rather thanspectral norm.7.2 Second prevalent problem:Projection on EDM cone in √ d ijLet◦√D [√dij ] ∈ K = S N h ∩ R N×N+ (1347)be an unknown matrix of absolute distance; id est,D = [d ij ] ◦√ D ◦ ◦√ D ∈ EDM N (1348)where ◦ denotes Hadamard product. The second prevalent proximityproblem is a Euclidean projection (in the natural coordinates √ d ij ) of matrixH on a nonconvex subset of the boundary of the nonconvex cone of Euclideanabsolute-distance matrices rel∂ √ EDM N : (6.3, confer Figure 138b)minimize ‖ ◦√ ⎫D − H‖◦√ 2 FD⎪⎬subject to rankVN TDV N ≤ ρ Problem 2 (1349)◦√ √D ∈ EDMN⎪⎭where√EDM N = { ◦√ D | D ∈ EDM N } (1182)This statement of the second proximity problem is considered difficult tosolve because of the constraint on desired affine dimension ρ (5.7.2) andbecause the objective function‖ ◦√ D − H‖ 2 F = ∑ i,j( √ d ij − h ij ) 2 (1350)is expressed in the natural coordinates; projection on a doubly nonconvexset.