v2007.09.13 - Convex Optimization

3.1. CONVEX FUNCTION 211[ Yt]∈ epif ⇔ t ≥ f(Y ) ⇒ [ ∇f(X) T−1 ]([ Yt] [−Xf(X)])≤ 0(511)This means, for each and every point X in the domain of a real convexfunction [ f(X) ] , there exists a hyperplane ∂H − [ in R p × ] R having normal∇f(X)Xsupporting the function epigraph at ∈ ∂H−1f(X)−{[ Y∂H − =t] [ Rp∈R][∇f(X)T−1 ]([ Yt] [−Xf(X)]) }= 0(512)One such supporting hyperplane (confer Figure 20(a)) is illustrated inFigure 59 for a convex quadratic.From (510) we deduce, for each and every X,Y ∈ domf∇f(X) T (Y − X) ≥ 0 ⇒ f(Y ) ≥ f(X) (513)meaning, the gradient at X identifies a supporting hyperplane there in R p{Y ∈ R p | ∇f(X) T (Y − X) = 0} (514)to the convex sublevel sets of convex function f (confer (462))L f(X) f ∆ = {Y ∈ domf | f(Y ) ≤ f(X)} ⊆ R p (515)illustrated for an arbitrary real convex function in Figure 60.3.1.10 first-order convexity condition, vector functionNow consider the first-order necessary and sufficient condition for convexityof a vector-valued function: Differentiable function f(X) : R p×k →R M isconvex if and only if domf is open, convex, and for each and everyX,Y ∈ domff(Y ) ≽R M +f(X) +→Y −Xdf(X)= f(X) + d dt∣ f(X+ t (Y − X)) (516)t=0