v2006.03.09 - Convex Optimization

204 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONSαβα ≥ β ≥ γ∇ ˆf(X)Y−Xγ{Z | ˆf(Z) = α}{Y | ∇ ˆf(X) T (Y − X) = 0, ˆf(X)=α}Figure 55: Shown is a plausible contour plot in R 2 of some arbitrary realconvex function ˆf(Z) at selected levels α , β , and γ ; contours of equallevel ˆf (level sets) drawn in the function’s domain. A convex functionhas convex sublevel sets L ˆf(X)ˆf (422). [194,4.6] 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. [123, 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 16 and Figure 51.