v2007.09.13 - Convex Optimization

212 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONSαβα ≥ β ≥ γ∇f(X)Y−Xγ{Z | f(Z) = α}{Y | ∇f(X) T (Y − X) = 0, f(X)=α}Figure 60: Shown is a plausible contour plot in R 2 of some arbitrary realconvex function f(Z) at selected levels α , β , and γ ; contours of equallevel f (level sets) drawn in the function’s domain. A convex functionhas convex sublevel sets L f(X) f (515). [228,4.6] The sublevel set whoseboundary is the level set at α , for instance, comprises all the shaded regions.For any particular convex function, the family comprising all its sublevel setsis nested. [147, p.75] Were the sublevel sets not convex, we may certainlyconclude the corresponding function is neither convex. Contour plots of realaffine functions are illustrated in Figure 17 and Figure 55.