v2007.09.13 - Convex Optimization

5.4. EDM DEFINITION 325because (A.3.1.0.5)Ω ≽ 0 ⇒ Θ T Θ ≽ 0 (778)Decomposition (775) and the relative-angle matrix inequality Ω ≽ 0 lead toa different expression of an inner-product form EDM definition (770)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)(779)and another expression of the EDM cone:EDM N ={D(Ω,d) | Ω ≽ 0, √ }δ(d) ≽ 0(780)In the particular circumstance x 1 = 0, we can relate interpoint anglematrix Ψ from the Gram decomposition in (714) to relative-angle matrixΩ in (775). Thus,[ ] 1 0TΨ ≡ , x0 Ω 1 = 0 (781)5.4.3.2 Inner-product form −V T N D(Θ)V N convexityOn[√page 323]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) (770) 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 (774)−V T N D(Θ)V N = Θ T Θ (782)is a convex quadratic function of Θ on domain R n×N−1 achieving its minimumat Θ = 0.