v2010.10.26 - Convex Optimization

3.4. AFFINE FUNCTION 239f(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 73: 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 planewith third dimension. We say sequence of hyperplanes, w.r.t domain R 2 ofaffine function f(z)= a T z + b : R 2 → R , is increasing in normal direction(Figure 26) because affine function increases in direction of gradient a(3.6.0.0.3). Minimization of a T z + b over C is equivalent to minimizationof a T z .