v2009.01.01 - Convex Optimization

196 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
3.1 Convex function
3.1.1 real and vector-valued function
Vector-valued function
f(X) : R p×k →R M =
⎡
⎢
⎣
f 1 (X)
.
f M (X)
⎤
⎥
⎦ (441)
assigns each X in its domain domf (a subset of ambient vector space R p×k )
to a specific element [222, p.3] of its range (a subset of R M ). Function f(X)
is linear in X on its domain if and only if, for each and every Y,Z ∈domf
and α , β ∈ R
f(αY + βZ) = αf(Y ) + βf(Z) (442)
A vector-valued function f(X) : R p×k →R M is convex in X if and only if
domf is a convex set and, for each and every Y,Z ∈domf and 0≤µ≤1
f(µY + (1 − µ)Z) ≼
µf(Y ) + (1 − µ)f(Z) (443)
R M +
As defined, continuity is implied but not differentiability. Some, but not
all, nonlinear functions are convex. Reversing sense of the inequality flips
this definition to concavity. Linear functions are, apparently, simultaneously
convex and concave.
Vector-valued functions are most often compared (164) as in (443) with
respect to the M-dimensional self-dual nonnegative orthant R M + , a proper
cone. 3.2 In this case, the test prescribed by (443) is simply a comparison
on R of each entry f i of a vector-valued function f . (2.13.4.2.3) The
vector-valued function case is therefore a straightforward generalization of
conventional convexity theory for a real function. This conclusion follows
from theory of dual generalized inequalities (2.13.2.0.1) which asserts
3.2 The definition of convexity can be broadened to other (not necessarily proper) cones;
referred to in the literature as K-convexity. [255]