v2009.01.01 - Convex Optimization

514 CHAPTER 7. PROXIMITY PROBLEMS
by (1585) where
{ ∣
µ ⋆ = max
λ ( −VN T (D ⋆ − H)V N
∣ , i = 1... N −1
i
)i
} ∈ R + (1251)
the minimized largest absolute eigenvalue (due to matrix symmetry).
For lack of a unique solution here, we prefer the Frobenius rather than
spectral norm.
7.2 Second prevalent problem:
Projection on EDM cone in √ d ij
Let
◦√
D ∆ = [ √ d ij ] ∈ K = S N h ∩ R N×N
+ (1252)
be an unknown matrix of absolute distance; id est,
D = [d ij ] ∆ = ◦√ D ◦ ◦√ D ∈ EDM N (1253)
where ◦ denotes Hadamard product. The second prevalent proximity
problem is a Euclidean projection (in the natural coordinates √ d ij ) of matrix
H on a nonconvex subset of the boundary of the nonconvex cone of Euclidean
absolute-distance matrices rel∂ √ EDM N : (6.3, confer Figure 114(b))
minimize ‖ ◦√ ⎫
D − H‖
◦√ 2 F
D
⎪⎬
subject to rankVN TDV N ≤ ρ Problem 2 (1254)
◦√ √
D ∈ EDM
N
⎪⎭
where
√
EDM N = { ◦√ D | D ∈ EDM N } (1087)
This statement of the second proximity problem is considered difficult to
solve because of the constraint on desired affine dimension ρ (5.7.2) and
because the objective function
‖ ◦√ D − H‖ 2 F = ∑ i,j
( √ d ij − h ij ) 2 (1255)
is expressed in the natural coordinates; projection on a doubly nonconvex
set.