v2004.06.19 - Convex Optimization

264 CHAPTER 7. EDM PROXIMITYFor the same reasons as before, y ij →h ij d ij as optimization proceeds.The similarity of problem (762) to (745) cannot go unnoticed, but thepossibility of numerical instability due to division by small numbers here istroublesome.7.3.1.1.1 Example. Alternating projection on nearest EDM.By solving (762) we confirm the result from an example given by Glunt,Hayden, et alii [136, §6] who found an analytical solution to (757) for theparticular cardinality N = 3 by using the alternating projection method ofvon Neumann (§E.9):⎡H = ⎣0 1 11 0 91 9 0⎤⎦ , D ⋆ =Let the alternate convex sets be the two conesK 1 = S N 0K 2 =⋂⎡⎢⎣{A ∈ S N | 〈yy T , −A〉 ≥ 0 }19 1909 919 7609 919 7609 9⎤⎥⎦ (764)y∈N(1 T )= { A ∈ S N | −y T V AV y ≥ 0, ∀yy T (≽ 0) } (765)= { A ∈ S N | −V AV ≽ 0 }where auxiliary matrix V ∈ S N is the geometric centering matrix (996), soK 1 ∩ K 2 = EDM N (766)The dual cone K ∗ 1= S N⊥0 ⊆ S N (51) is an orthogonal subspace, and from thedual EDM cone development in §5.6.1,K ∗ 2 = − cone { V N υυ T V T N | υ ∈ R N−1} ⊂ S N (767)In [137, §5.3] Gaffke & Mathar observe that projection on K 1 and K 2 havesimple closed forms: Projection on subspace K 1 is easily performed by symmetrizationand zeroing the main diagonal or vice versa, while projection ofH ∈ S N on K 2 isP K2 H = H − P S N+(V H V ) (768)Thus the original problem (757) of projecting H on the EDM cone is transformed,basically, to an equivalent sequence of projections on the PSD cone.