v2009.01.01 - Convex Optimization
v2009.01.01 - Convex Optimization v2009.01.01 - Convex Optimization
486 CHAPTER 6. CONE OF DISTANCE MATRICES D ◦ = δ(D ◦ 1) + (D ◦ − δ(D ◦ 1)) ∈ S N⊥ h ⊕ S N c ∩ S N + = EDM N◦ S N c ∩ S N + EDM N◦ D ◦ − δ(D ◦ 1) D ◦ δ(D ◦ 1) 0 EDM N◦ S N⊥ h Figure 126: Hand-drawn abstraction of polar EDM cone (drawn truncated). Any member D ◦ of polar EDM cone can be decomposed into two linearly independent nonorthogonal components: δ(D ◦ 1) and D ◦ − δ(D ◦ 1).
6.8. DUAL EDM CONE 487 has dual affine dimension complementary to affine dimension corresponding to the optimal solution of minimize ‖D − H‖ F D∈S N h subject to −VN TDV N ≽ 0 (1189) Precisely, rank(D ◦⋆ −δ(D ◦⋆ 1)) + rank(V T ND ⋆ V N ) = N −1 (1190) and rank(D ◦⋆ −δ(D ◦⋆ 1))≤N−1 because vector 1 is always in the nullspace of rank’s argument. This is similar to the known result for projection on the self-dual positive semidefinite cone and its polar: rankP −S N + H + rankP S N + H = N (1191) When low affine dimension is a desirable result of projection on the EDM cone, projection on the polar EDM cone should be performed instead. Convex polar problem (1188) can be solved for D ◦⋆ by transforming to an equivalent Schur-form semidefinite program (3.1.7.2). Interior-point methods for numerically solving semidefinite programs tend to produce high-rank solutions. (4.1.1.1) Then D ⋆ = H − D ◦⋆ ∈ EDM N by Corollary E.9.2.2.1, and D ⋆ will tend to have low affine dimension. This approach breaks when attempting projection on a cone subset discriminated by affine dimension or rank, because then we have no complementarity relation like (1190) or (1191) (7.1.4.1). Lemma 6.8.1.1.1; rewriting, minimize ‖(D ◦ − δ(D ◦ 1)) − (H − δ(D ◦ 1))‖ F D ◦ ∈ S N subject to D ◦ − δ(D ◦ 1) ≽ 0 which is the projection of affinely transformed optimal solution H − δ(D ◦⋆ 1) on S N c ∩ S N + ; D ◦⋆ − δ(D ◦⋆ 1) = P S N + P S N c (H − δ(D ◦⋆ 1)) Foreknowledge of an optimal solution D ◦⋆ as argument to projection suggests recursion.
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486 CHAPTER 6. CONE OF DISTANCE MATRICES<br />
D ◦ = δ(D ◦ 1) + (D ◦ − δ(D ◦ 1)) ∈ S N⊥<br />
h<br />
⊕ S N c ∩ S N + = EDM N◦<br />
S N c ∩ S N +<br />
EDM N◦ D ◦ − δ(D ◦ 1)<br />
D ◦<br />
δ(D ◦ 1)<br />
0<br />
EDM N◦<br />
S N⊥<br />
h<br />
Figure 126: Hand-drawn abstraction of polar EDM cone (drawn truncated).<br />
Any member D ◦ of polar EDM cone can be decomposed into two linearly<br />
independent nonorthogonal components: δ(D ◦ 1) and D ◦ − δ(D ◦ 1).
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