v2010.10.26 - Convex Optimization

204 CHAPTER 2. CONVEX GEOMETRYαα ≥ β ≥ γβCx ⋆ −∇f(x ⋆ )γ{z | f(z) = α}{y | ∇f(x ⋆ ) T (y − x ⋆ ) = 0, f(x ⋆ )=γ}Figure 67: (confer Figure 78) Shown is a plausible contour plot in R 2 ofsome arbitrary differentiable convex real function f(x) at selected levels α ,β , and γ ; id est, contours of equal level f (level sets) drawn dashed infunction’s domain. From results in3.6.2 (p.258), gradient ∇f(x ⋆ ) is normalto γ-sublevel set L γ f (559) by Definition E.9.1.0.1. From2.13.10.1, functionis minimized over convex set C at point x ⋆ iff negative gradient −∇f(x ⋆ )belongs to normal cone to C there. In circumstance depicted, normal coneis a ray whose direction is coincident with negative gradient. So, gradient isnormal to a hyperplane supporting both C and the γ-sublevel set.