v2009.01.01 - Convex Optimization

3.1. CONVEX FUNCTION 209
rather
]
x ≥ 0, y ≥ x 1/α
⇔
t i
[
ti−1 t i
y b i
≽ 0 ,
i=1... q
(487)
x = t q ≥ 0
3.1.6 affine function
A function f(X) is affine when it is continuous and has the dimensionally
extensible form (confer2.9.1.0.2)
f(X) = AX + B (488)
When B=0 then f(X) is a linear function. All affine functions are
simultaneously convex and concave. Affine multidimensional functions are
recognized by existence of no multivariate terms (multiplicative in argument
entries) and no polynomial terms of degree higher than 1 ; id est, entries
of the function are characterized only by linear combinations of argument
entries plus constants.
The real affine function in Figure 63 illustrates hyperplanes, in its
domain, constituting contours of equal function-value (level sets (492)).
3.1.6.0.1 Example. Engineering control. [339,2.2] 3.10
For X ∈ S M and matrices A,B, Q, R of any compatible dimensions, for
example, the expression XAX is not affine in X whereas
g(X) =
[ R B T X
XB Q + A T X + XA
]
(489)
is an affine multidimensional function.
engineering control. [51] [126]
Such a function is typical in
Any single- or many-valued inverse of an affine function is affine.
3.10 The interpretation from this citation of {X ∈ S M | g(X) ≽ 0} as “an intersection
between a linear subspace and the cone of positive semidefinite matrices” is incorrect.
(See2.9.1.0.2 for a similar example.) The conditions they state under which strong
duality holds for semidefinite programming are conservative. (confer4.2.3.0.1)