v2007.09.13 - Convex Optimization

3.1. CONVEX FUNCTION 193f(z)Az 2z 1CH−H+a{z ∈ R 2 | a T z = κ1 }{z ∈ R 2 | a T z = κ2 }{z ∈ R 2 | a T z = κ3 }Figure 55: Cartesian axes in R 3 and three hyperplanes intersecting convex setC ⊂ R 2 reproduced from Figure 19. Plotted with third dimension is affine setA = f(R 2 ) a plane. Sequence of hyperplanes, w.r.t domain R 2 of an affinefunction f(z)= a T z + b : R 2 → R , is increasing in direction of gradient a(3.1.8.0.3) because affine function increases in normal direction (Figure 17).is an affine multidimensional function.engineering control. [293,2.2] 3.5 [44] [101]Such a function is typical in3.1.6.0.1 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 55. Sincevector b is fixed, the problem posed is the same as the convex optimizationminimize a T zzsubject to z ∈ C(457)3.5 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)