v2009.01.01 - Convex Optimization

234 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
3.2.0.0.2 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 (68)
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
(559)
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) (560)
3.2.0.0.3 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 (560).
3.2.1 first-order convexity condition, matrix function
From the scalar-definition (3.2.0.0.1) 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 (561)
that follows immediately from the first-order condition (544) for convexity of
a real function because
→Y −X
T
w dg(X) w = 〈 ∇ X w T g(X)w , Y − X 〉 (562)