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

3.8. MATRIX-VALUED CONVEX FUNCTION 253
Given symmetric nonnegative data [h ij ] ∈ S N ∩ R N×N
+ , consider function
f(vec X) =
N−1
∑
i=1 j=i+1
N∑
(|x i − x j | − h ij ) 2 ∈ R (1284)
Take the gradient and Hessian of f . Then explain why f is not a convex
function; id est, why doesn’t second-order condition (595) apply to the
constant positive semidefinite Hessian matrix you found. For N = 6 and h ij
data from (1357), apply line theorem 3.8.3.0.1 to plot f along some arbitrary
lines through its domain.
3.7.4.1 second-order ⇒ first-order condition
For a twice-differentiable real function f i (X) : R p →R having open domain,
a consequence of the mean value theorem from calculus allows compression
of its complete Taylor series expansion about X ∈ domf i (D.1.7) to three
terms: On some open interval of ‖Y ‖ 2 , so that each and every line segment
[X,Y ] belongs to domf i , there exists an α∈[0, 1] such that [384,1.2.3]
[41,1.1.4]
f i (Y ) = f i (X) + ∇f i (X) T (Y −X) + 1 2 (Y −X)T ∇ 2 f i (αX + (1 − α)Y )(Y −X)
(596)
The first-order condition for convexity (586) follows directly from this and
the second-order condition (595).
3.8 Matrix-valued convex function
We need different tools for matrix argument: We are primarily interested in
continuous matrix-valued functions g(X). We choose symmetric g(X)∈ S M
because matrix-valued functions are most often compared (597) with respect
to the positive semidefinite cone S M + in the ambient space of symmetric
matrices. 3.20
3.20 Function symmetry is not a necessary requirement for convexity; indeed, for A∈R m×p
and B ∈R m×k , g(X) = AX + B is a convex (affine) function in X on domain R p×k with
respect to the nonnegative orthant R m×k
+ . Symmetric convex functions share the same
benefits as symmetric matrices. Horn & Johnson [198,7.7] liken symmetric matrices to
real numbers, and (symmetric) positive definite matrices to positive real numbers.