v2009.01.01 - Convex Optimization

354 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
5.4.1 −V T N D(X)V N convexity
([ ])
xi
We saw that EDM entries d ij are convex quadratic functions. Yet
x j
−D(X) (798) is not a quasiconvex function of matrix X ∈ R n×N because the
second directional derivative (3.3)
− d2
dt 2 ∣
∣∣∣t=0
D(X+ t Y ) = 2 ( −δ(Y T Y )1 T − 1δ(Y T Y ) T + 2Y T Y ) (802)
is indefinite for any Y ∈ R n×N since its main diagonal is 0. [134,4.2.8]
[176,7.1, 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 (803)
where we introduce the full-rank skinny Schoenberg auxiliary matrix (B.4.2)
⎡
V ∆ N = √ 1
2 ⎢
⎣
(N(V N )=0) having range
−1 −1 · · · −1
1 0
1
. . .
0 1
⎤
⎥
⎦
= 1 √
2
[ −1
T
I
]
∈ R N×N−1 (804)
R(V N ) = N(1 T ) , V T N 1 = 0 (805)
Matrix-valued function (803) meets the criterion for convexity in3.2.3.0.2
over its domain that is all of R n×N ; videlicet, for any Y ∈ R n×N
− d2
dt 2 V T N D(X + t Y )V N = 4V T N Y T Y V N ≽ 0 (806)
Quadratic matrix-valued function −VN TD(X)V N is therefore convex in X
achieving its minimum, with respect to a positive semidefinite cone (2.7.2.2),
at X = 0. When the penultimate number of points exceeds the dimension
of the space n < N −1, strict convexity of the quadratic (803) becomes
impossible because (806) could not then be positive definite.