Chapter 3 Geometry of convex functions - Meboo Publishing ...

254 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
3.8.0.0.1 Definition. Convex matrix-valued function:
1) Matrix-definition.
A function g(X) : R p×k →S M is convex in X iff domg is a convex set and,
for each and every Y,Z ∈domg and all 0≤µ≤1 [210,2.3.7]
g(µY + (1 − µ)Z) ≼
µg(Y ) + (1 − µ)g(Z) (597)
S M +
Reversing sense of the inequality flips this definition to concavity. Strict
convexity is defined less a stroke of the pen in (597) similarly to (478).
2) Scalar-definition.
It follows that g(X) : R p×k →S M is convex in X iff w T g(X)w : R p×k →R is
convex in X for each and every ‖w‖= 1; shown by substituting the defining
inequality (597). By dual generalized inequalities we have the equivalent but
more broad criterion, (2.13.5)
g convex w.r.t S M + ⇔ 〈W , g〉 convex
for each and every W ≽
S M∗
+
0 (598)
Strict convexity on both sides requires caveat W ≠ 0. Because the
set of all extreme directions for the selfdual positive semidefinite cone
(2.9.2.4) comprises a minimal set of generators for that cone, discretization
(2.13.4.2.1) allows replacement of matrix W with symmetric dyad ww T as
proposed.
△
3.8.0.0.2 Example. Taxicab distance matrix.
Consider an n-dimensional vector space R n with metric induced by the
1-norm. Then distance between points x 1 and x 2 is the norm of their
difference: ‖x 1 −x 2 ‖ 1 . Given a list of points arranged columnar in a matrix
X = [x 1 · · · x N ] ∈ R n×N (76)
then we could define a taxicab distance matrix
D 1 (X) (I ⊗ 1 T n) | vec(X)1 T − 1 ⊗X | ∈ S N h ∩ R N×N
+
⎡
⎤
0 ‖x 1 − x 2 ‖ 1 ‖x 1 − x 3 ‖ 1 · · · ‖x 1 − x N ‖ 1
‖x 1 − x 2 ‖ 1 0 ‖x 2 − x 3 ‖ 1 · · · ‖x 2 − x N ‖ 1
=
‖x
⎢ 1 − x 3 ‖ 1 ‖x 2 − x 3 ‖ 1 0 ‖x 3 − x N ‖ 1
⎥
⎣ . .
... . ⎦
‖x 1 − x N ‖ 1 ‖x 2 − x N ‖ 1 ‖x 3 − x N ‖ 1 · · · 0
(599)