v2010.10.26 - Convex Optimization

258 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONSWhen f(X) : R p →R is a real differentiable convex function with vectorargument on open convex domain, there is simplification of the first-ordercondition (607); for each and every X,Y ∈ domff(Y ) ≥ f(X) + ∇f(X) T (Y − X) (608)From this we can find ∂H − a unique [371, p.220-229] nonvertical [199,B.1.2]hyperplane [ ](2.4), expressed in terms of function gradient, supporting epifXat : videlicet, defining f(Y /∈ domf) ∞ [61,3.1.7]f(X)[ Yt]∈ epif ⇔ t ≥ f(Y ) ⇒ [ ∇f(X) T−1 ]([ Yt]−[ ]) X≤ 0f(X)(609)This means, for each and every point X in the domain of a convex real[ function ] 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]−[ ]) } X= 0f(X)(610)Such a hyperplane is strictly supporting whenever a function is strictlyconvex. One such supporting hyperplane (confer Figure 29a) is illustratedin Figure 77 for a convex quadratic.From (608) we deduce, for each and every X,Y ∈ domf in the domain,∇f(X) T (Y − X) ≥ 0 ⇒ f(Y ) ≥ f(X) (611)meaning, the gradient at X identifies a supporting hyperplane there in R p{Y ∈ R p | ∇f(X) T (Y − X) = 0} (612)to the convex sublevel sets of convex function f (confer (559))L f(X) f {Z ∈ domf | f(Z) ≤ f(X)} ⊆ R p (613)illustrated for an arbitrary convex real function in Figure 78 and Figure 67.That supporting hyperplane is unique for twice differentiable f . [219, p.501]