v2010.10.26 - Convex Optimization

3.6. GRADIENT 259αβα ≥ β ≥ γ∇f(X)Y−Xγ{Z | f(Z) = α}{Y | ∇f(X) T (Y − X) = 0, f(X)=α} (612)Figure 78:(confer Figure 67) Shown is a plausible contour plot in R 2 of somearbitrary real differentiable 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 (613). [307,4.6] Thesublevel set whose boundary is the level set at α , for instance, comprisesall the shaded regions. For any particular convex function, the familycomprising all its sublevel sets is nested. [199, p.75] Were sublevel sets notconvex, we may certainly conclude the corresponding function is neitherconvex. Contour plots of real affine functions are illustrated in Figure 26and Figure 73.