v2006.03.09 - Convex Optimization

3.1. CONVEX FUNCTION 195Variegated multidimensional affine functions are recognized by theexistence of no multivariate terms in argument entries and no polynomialterms in argument entries of degree higher than 1 ; id est, entries of thefunction are characterized only by linear combinations of the argumententries plus constants.For 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 (402)X + XAis an affine multidimensional function.engineering control. [256,2.2] 3.4 [37] [81]Such a function is typical in3.1.1.2.1 Example. Linear objective.Consider minimization of a real affine function ˆf(z)= a T z + b over theconvex feasible set C in its domain R 2 illustrated in Figure 51. Sincevector b is fixed, the problem posed is the same as the convex optimizationminimize a T zzsubject to z ∈ C(403)whose objective of minimization is a real linear function. Were convex set Cpolyhedral (2.12), then this problem would be called a linear program.Were C a positive semidefinite cone, then this problem would be called asemidefinite program.There are two distinct ways to visualize this problem: one in theobjective function’s domain R 2 , [ the other ] including the ambient space of theR2objective function’s range as in . Both visualizations are illustratedRin Figure 51. Visualization in the function domain is easier because of lowerdimension and because level sets of any affine function are affine (2.1.9).In this circumstance, the level sets are parallel hyperplanes with respectto R 2 . One solves optimization problem (403) graphically by finding thathyperplane intersecting feasible set C furthest right (in the direction ofnegative gradient (3.1.1.4)).3.4 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. (confer6.2.3.0.1)