v2007.09.17 - Convex Optimization

5.4. EDM DEFINITION 327because (A.3.1.0.5)Ω ≽ 0 ⇒ Θ T Θ ≽ 0 (782)Decomposition (779) and the relative-angle matrix inequality Ω ≽ 0 lead toa different expression of an inner-product form EDM definition (774)D(Ω,d) ∆ ==[ ] 01dT + 1 [ 0 d ] √ ([ ]) [ ] √ ([ ])0 0 0 T T 0− 2 δδd 0 Ω d[ ]0 dTd d1 T + 1d T − 2 √ δ(d) Ω √ ∈ EDM Nδ(d)(783)and another expression of the EDM cone:EDM N ={D(Ω,d) | Ω ≽ 0, √ }δ(d) ≽ 0(784)In the particular circumstance x 1 = 0, we can relate interpoint anglematrix Ψ from the Gram decomposition in (718) to relative-angle matrixΩ in (779). Thus,[ ] 1 0TΨ ≡ , x0 Ω 1 = 0 (785)5.4.3.2 Inner-product form −V T N D(Θ)V N convexityOn[√page 325]we saw that each EDM entry d ij is a convex quadratic functiondikof √dkj and a quasiconvex function of θ ikj . Here the situation forinner-product form EDM operator D(Θ) (774) is identical to that in5.4.1for list-form D(X) ; −D(Θ) is not a quasiconvex function of Θ by the samereasoning, and from (778)−V T N D(Θ)V N = Θ T Θ (786)is a convex quadratic function of Θ on domain R n×N−1 achieving its minimumat Θ = 0.