v2010.10.26 - Convex Optimization

238 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONSdefinition (497). All affine functions satisfy a membership relation, forsome normal cone, like (506). Affine multidimensional functions are moreeasily recognized by existence of no multiplicative multivariate terms and nopolynomial terms of degree higher than 1 ; id est, entries of the function arecharacterized by only linear combinations of argument entries plus constants.A T XA + B T B is affine in X , for example. Trace is an affine function;actually, a real linear function expressible as inner product f(X) = 〈A, X〉with matrix A being the identity. The real affine function in Figure 73illustrates hyperplanes, in its domain, constituting contours of equalfunction-value (level sets (555)).3.4.0.0.1 Example. Engineering control. [388,2.2] 3.14For X ∈ S M and matrices A,B, Q, R of any compatible dimensions, forexample, the expression XAX is not affine in X whereas[ ]R Bg(X) =T XXB Q + A T (551)X + XAis an affine multidimensional function.engineering control. [59] [151]Such a function is typical in(confer Figure 16) Any single- or many-valued inverse of an affine functionis affine.3.4.0.0.2 Example. Linear objective.Consider minimization of a real affine function f(z)= a T z + b over theconvex feasible set C in its domain R 2 illustrated in Figure 73. Since scalar bis fixed, the problem posed is the same as the convex optimizationminimize a T zzsubject to z ∈ C(552)whose objective of minimization is a real linear function. Were convex set Cpolyhedral (2.12), then this problem would be called a linear program. Were3.14 The interpretation from this citation of {X ∈ S M | g(X) ≽ 0} as “an intersectionbetween 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 strongduality holds for semidefinite programming are conservative. (confer4.2.3.0.1)