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

3.9. QUASICONVEX 261
3.9.0.0.2 Definition. Quasiconvex multidimensional function.
Scalar-, vector-, or matrix-valued function g(X) : R p×k →S M is a quasiconvex
function of matrix X iff domg is a convex set and the sublevel set
corresponding to each and every S ∈ S M
L S
g = {X ∈ dom g | g(X) ≼ S } ⊆ R p×k (607)
is convex. Vectors are compared with respect to the nonnegative orthant R M +
while matrices are with respect to the positive semidefinite cone S M + .
Convexity of the superlevel set corresponding to each and every S ∈ S M ,
likewise
L S g = {X ∈ domg | g(X) ≽ S } ⊆ R p×k (620)
is necessary and sufficient for quasiconcavity.
3.9.0.0.3 Exercise. Nonconvexity of matrix product.
Consider real function f on a positive definite domain
[ ] [
X1
rel int S
N
+
f(X) = tr(X 1 X 2 ) , X ∈ domf
X 2 rel int S N +
with superlevel sets
]
△
(621)
L s f = {X ∈ dom f | 〈X 1 , X 2 〉 ≥ s } (622)
Prove: f(X) is not quasiconcave except when N = 1, nor is it quasiconvex
unless X 1 = X 2 .
3.9.1 bilinear
Bilinear function 3.24 x T y of vectors x and y is quasiconcave (monotonic) on
the entirety of the nonnegative orthants only when vectors are of dimension 1.
3.9.2 quasilinear
When a function is simultaneously quasiconvex and quasiconcave, it is called
quasilinear. Quasilinear functions are completely determined by convex
level sets. One-dimensional function f(x) = x 3 and vector-valued signum
function sgn(x) , for example, are quasilinear. Any monotonic function is
quasilinear 3.25 (but not vice versa, Exercise 3.7.4.0.1).
3.24 Convex envelope of bilinear functions is well known. [3]
3.25 e.g., a monotonic concave function is quasiconvex, but dare not confuse these terms.