v2009.01.01 - Convex Optimization

230 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
α
β
α ≥ β ≥ γ
∇f(X)
Y−X
γ
{Z | f(Z) = α}
{Y | ∇f(X) T (Y − X) = 0, f(X)=α}
Figure 68: Shown is a plausible contour plot in R 2 of some arbitrary real
convex function f(Z) at selected levels α , β , and γ ; contours of equal
level f (level sets) drawn in the function’s domain. A convex function
has convex sublevel sets L f(X) f (550). [266,4.6] The sublevel set whose
boundary is the level set at α , for instance, comprises all the shaded regions.
For any particular convex function, the family comprising all its sublevel sets
is nested. [173, p.75] Were the sublevel sets not convex, we may certainly
conclude the corresponding function is neither convex. Contour plots of real
affine functions are illustrated in Figure 22 and Figure 63.