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

3.8. MATRIX-VALUED CONVEX FUNCTION 255
where 1 n is a vector of ones having dim1 n = n and where ⊗ represents
Kronecker product. This matrix-valued function is convex with respect to the
nonnegative orthant since, for each and every Y,Z ∈ R n×N and all 0≤µ≤1
D 1 (µY + (1 − µ)Z)
≼
R N×N
+
µD 1 (Y ) + (1 − µ)D 1 (Z) (600)
3.8.0.0.3 Exercise. 1-norm distance matrix.
The 1-norm is called taxicab distance because to go from one point to another
in a city by car, road distance is a sum of grid lengths. Prove (600).
3.8.1 first-order convexity condition, matrix function
From the scalar-definition (3.8.0.0.1) of a convex matrix-valued function,
for differentiable function g and for each and every real vector w of unit
norm ‖w‖= 1, we have
w T g(Y )w ≥ w T →Y −X
T
g(X)w + w dg(X) w (601)
that follows immediately from the first-order condition (585) for convexity of
a real function because
→Y −X
T
w dg(X) w = 〈 ∇ X w T g(X)w , Y − X 〉 (602)
→Y −X
where dg(X) is the directional derivative (D.1.4) of function g at X in
direction Y −X . By discretized dual generalized inequalities, (2.13.5)
g(Y ) − g(X) −
→Y −X
dg(X) ≽
S M +
0 ⇔
〈
g(Y ) − g(X) −
For each and every X,Y ∈ domg (confer (592))
→Y −X
〉
dg(X) , ww T ≥ 0 ∀ww T (≽ 0)
(603)
S M∗
+
g(Y ) ≽
g(X) +
→Y −X
dg(X) (604)
S M +
must therefore be necessary and sufficient for convexity of a matrix-valued
function of matrix variable on open convex domain.