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

212 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)
⎤
⎥
⎦ (473)
assigns each X in its domain domf (a subset of ambient vector space R p×k )
to a specific element [252, 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) (474)
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) (475)
R M +
As defined, continuity is implied but not differentiability (nor smoothness). 3.2
Apparently some, but not all, nonlinear functions are convex. Reversing sense
of the inequality flips this definition to concavity. Linear (and affine3.5)
functions attain equality in this definition. Linear functions are therefore
simultaneously convex and concave.
Vector-valued functions are most often compared (173) as in (475) with
respect to the M-dimensional selfdual nonnegative orthant R M + , a proper
cone. 3.3 In this case, the test prescribed by (475) 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
f convex w.r.t R M + ⇔ w T f convex ∀w ∈ G(R M∗
+ ) (476)
3.2 Figure 66b illustrates a nondifferentiable convex function. Differentiability is certainly
not a requirement for optimization of convex functions by numerical methods; e.g., [238].
3.3 Definition of convexity can be broadened to other (not necessarily proper) cones.
Referred to in the literature as K-convexity, [289] R M∗
+ (476) generalizes to K ∗ .