v2007.11.26 - Convex Optimization

314 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX5.4.1 −V T N D(X)V N convexity([ ])xiWe saw that EDM entries d ij are convex quadratic functions. Yetx j−D(X) (734) is not a quasiconvex function of matrix X ∈ R n×N because thesecond directional derivative (3.3)− d2dt 2 ∣∣∣∣t=0D(X+ t Y ) = 2 ( −δ(Y T Y )1 T − 1δ(Y T Y ) T + 2Y T Y ) (738)is indefinite for any Y ∈ R n×N since its main diagonal is 0. [110,4.2.8][150,7.1, prob.2] Hence −D(X) can neither be convex in X .The outcome is different when instead we consider−V T N D(X)V N = 2V T NX T XV N (739)where we introduce the full-rank skinny Schoenberg auxiliary matrix (B.4.2)⎡V ∆ N = √ 12 ⎢⎣(N(V N )=0) having range−1 −1 · · · −11 01. . .0 1⎤⎥⎦= 1 √2[ −1TI]∈ R N×N−1 (740)R(V N ) = N(1 T ) , V T N 1 = 0 (741)Matrix-valued function (739) meets the criterion for convexity in3.2.3.0.2over its domain that is all of R n×N ; videlicet, for any Y ∈ R n×N− d2dt 2 V T N D(X + t Y )V N = 4V T N Y T Y V N ≽ 0 (742)Quadratic matrix-valued function −VN TD(X)V N is therefore convex in Xachieving its minimum, with respect to a positive semidefinite cone (2.7.2.2),at X = 0. When the penultimate number of points exceeds the dimensionof the space n < N −1, strict convexity of the quadratic (739) becomesimpossible because (742) could not then be positive definite.