v2007.09.13 - Convex Optimization

456 CHAPTER 7. PROXIMITY PROBLEMSwhere{ ∣µ ⋆ = maxλ ( −VN T (D ⋆ − H)V N∣ , i = 1... N −1i)i} ∈ R + (1148)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 ∆ = [ √ d ij ] ∈ K = S N h ∩ R N×N+ (1149)be an unknown matrix of absolute distance; id est,D = [d ij ] ∆ = ◦√ D ◦ ◦√ D ∈ EDM N (1150)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 94(b))minimize ‖ ◦√ ⎫D − H‖◦√ 2 FD⎪⎬subject to rankVN TDV N ≤ ρ Problem 2 (1151)◦√ √D ∈ EDMN⎪⎭where√EDM N = { ◦√ D | D ∈ EDM N } (985)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 (1152)is expressed in the natural coordinates; projection on a doubly nonconvexset.