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

230 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
f(z)
A
z 2
z 1
C
H−
H+
a
{z ∈ R 2 | a T z = κ1 }
{z ∈ R 2 | a T z = κ2 }
{z ∈ R 2 | a T z = κ3 }
Figure 71: Three hyperplanes intersecting convex set C ⊂ R 2 from Figure 28.
Cartesian axes in R 3 : Plotted is affine subset A = f(R 2 )⊂ R 2 × R ; a plane
with third dimension. Sequence of hyperplanes, w.r.t domain R 2 of affine
function f(z)= a T z + b : R 2 → R , is increasing in direction of gradient a
(3.7.0.0.3) because affine function increases in normal direction (Figure 26).
Minimization of a T z + b over C is equivalent to minimization of a T z .
3.5.0.0.1 Example. Engineering control. [381,2.2] 3.13
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
[ ]
R B
g(X) =
T X
XB Q + A T (529)
X + XA
is an affine multidimensional function.
engineering control. [57] [147]
Such a function is typical in
(confer Figure 16) Any single- or many-valued inverse of an affine function
is affine.
3.13 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)