v2010.10.26 - Convex Optimization

5.4. EDM DEFINITION 431where ([√the numerator ]) forms an inner product of vectors. Distance-squaredikd ij √dkj is a convex quadratic function 5.23 on R 2 + whereas d ij (θ ikj ) isquasiconvex (3.8) minimized over domain {−π ≤θ ikj ≤π} by θ ⋆ ikj =0, weget the Pythagorean theorem when θ ikj = ±π/2, and d ij (θ ikj ) is maximizedwhen θ ⋆ ikj =±π ; d ij = (√ d ik + √ ) 2,d kj θikj = ±πsod ij = d ik + d kj , θ ikj = ± π 2d ij = (√ d ik − √ d kj) 2,θikj = 0(955)| √ d ik − √ d kj | ≤ √ d ij ≤ √ d ik + √ d kj (956)Hence the triangle inequality, Euclidean metric property 4, holds for anyEDM D .We may construct an inner-product form of the EDM definition formatrices by evaluating (953) for k=1 : By defining⎡√ √ √ ⎤d 12 d12 d 13 cos θ 213 d12 d 14 cos θ 214 · · · d12 d 1N cos θ 21N√ √ √ d12 d 13 cos θ 213 d 13 d13 d 14 cos θ 314 · · · d13 d 1N cos θ 31NΘ T Θ √ √ √ d12 d 14 cos θ 214 d13 d 14 cos θ 314 d 14... d14 d 1N cos θ 41N∈ S N−1⎢⎥⎣ ........√ √. ⎦√d12 d 1N cos θ 21N d13 d 1N cos θ 31N d14 d 1N cos θ 41N · · · d 1Nthen any EDM may be expressed[ ]0D(Θ) δ(Θ T 1Θ)T + 1 [ [0 δ(Θ T Θ) T] 0 0T− 20 Θ T Θ=[0 δ(Θ T Θ) Tδ(Θ T Θ) δ(Θ T Θ)1 T + 1δ(Θ T Θ) T − 2Θ T Θ]](957)∈ EDM N(958)EDM N = { D(Θ) | Θ ∈ R N−1×N−1} (959)for which all Euclidean metric properties hold. Entries of Θ T Θ result fromvector inner-products as in (954); id est,[]5.23 1 −e ıθ ikj−e −ıθ ≽ 0, having eigenvalues {0,2}. Minimum is attained forikj1[ √ ] {dik√dkjµ1, µ ≥ 0, θikj = 0=0, −π ≤ θ ikj ≤ π , θ ikj ≠ 0 . (D.2.1, [61, exmp.4.5])