v2007.11.26 - Convex Optimization

176 CHAPTER 2. CONVEX GEOMETRYαα ≥ β ≥ γβCx ⋆ −∇f(x ⋆ )γ{z | f(z) = α}{y | ∇f(x ⋆ ) T (y − x ⋆ ) = 0, f(x ⋆ )=γ}Figure 55: Shown is a plausible contour plot in R 2 of some arbitrarydifferentiable real convex function f(x) at selected levels α , β , and γ ;id est, contours of equal level f (level sets) drawn (dashed) in function’sdomain. Function is minimized over convex set C at point x ⋆ iff negativegradient −∇f(x ⋆ ) belongs to normal cone to C there. In circumstancedepicted, normal cone is a ray whose direction is coincident with negativegradient. From results in3.1.9 (p.212), ∇f(x ⋆ ) is normal to the γ-sublevelset by Definition E.9.1.0.1.a closed convex cone called the normal cone to K at point a . (E.10.3.2.1)From this, a new membership relation like (277) for closed convex cone K :y ∈ −(K − a) ∗ ⇔ 〈y , x − a〉≤0 for all x ∈ K (392)2.13.10.1 first-order optimality condition - restatementThe general first-order necessary and sufficient condition for optimalityof solution x ⋆ to a minimization problem with real differentiable convexobjective function f(x) : R n →R over convex feasible set C is [231,3]