v2009.01.01 - Convex Optimization

3.3. QUASICONVEX 241
Unlike convex functions, quasiconvex functions are not necessarily
continuous; e.g., quasiconcave rank(X) on S M + (2.9.2.6.2) and card(x)
on R M + . Although insufficient for convex functions, convexity of each and
every sublevel set serves as a definition of quasiconvexity:
3.3.0.0.2 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 dom g is a convex set and the sublevel set
corresponding to each and every S ∈ S M
L S
g = {X ∈ domg | g(X) ≼ S } ⊆ R p×k (567)
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 (579)
is necessary and sufficient for quasiconcavity.
△
3.3.0.0.3 Exercise. Nonconvexity of matrix product.
Consider real function f on a positive definite domain
f(X) = tr(X 1 X 2 ) , X ∆ =
[
X1
X 2
]
∈ domf ∆ =
[
rel int S
N
+
rel int S N +
]
(580)
with superlevel sets
L s f = {X ∈ domf | 〈X 1 , X 2 〉 ≥ s } (581)
Prove: f(X) is not quasiconcave except when N = 1, nor is it quasiconvex
unless X 1 = X 2 .
3.3.1 bilinear
Bilinear function 3.20 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.
3.20 Convex envelope of bilinear functions is well known. [2]