v2007.09.13 - Convex Optimization

3.1. CONVEX FUNCTION 207This equivalence to a hyperplane holds only for real functions. 3.8[ The ] Rpepigraph of the real affine function f(x) is therefore a halfspace in ,Rso we have:The real affine function is to convex functionsasthe hyperplane is to convex sets.Similarly, the matrix-valued affine function of real variable x , for anyparticular matrix A∈ R M×N ,describes a line in R M×N in direction Aand describes a line in R×R M×N{[h(x) : R→R M×N = Ax + B (504){Ax + B | x∈ R} ⊆ R M×N (505)xAx + B]}| x∈ R ⊂ R×R M×N (506)whose slope with respect to x is A .3.8 To prove that, consider a vector-valued affine functionf(x) : R p →R M = Ax + bhaving gradient ∇f(x)=A T ∈ R p×M : The affine set{[ ] }x| x∈ R p ⊂ R p ×R MAx + bis perpendicular toη ∆ =[∇f(x)−I]∈ R p×M × R M×Mbecauseη T ([xAx + b] [0−b])= 0 ∀x ∈ R pYet η is a vector (in R p ×R M ) only when M = 1.