v2009.01.01 - Convex Optimization
v2009.01.01 - Convex Optimization v2009.01.01 - Convex Optimization
182 CHAPTER 2. CONVEX GEOMETRY 2.13.10 Dual cone-translate First-order optimality condition (319) inspires a dual-cone variant: For any set K , the negative dual of its translation by any a∈ R n is −(K − a) ∗ ∆ = { y ∈ R n | 〈y , x − a〉≤0 for all x ∈ K } = { y ∈ R n | 〈y , x〉≤0 for all x ∈ K − a } (403) 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 (288) for closed convex cone K : y ∈ −(K − a) ∗ ⇔ 〈y , x − a〉≤0 for all x ∈ K (404) 2.13.10.1 first-order optimality condition - restatement The general first-order necessary and sufficient condition for optimality of solution x ⋆ to a minimization problem with real differentiable convex objective function f(x) : R n →R over convex feasible set C is [265,3] (confer (319)) −∇f(x ⋆ ) ∈ −(C − x ⋆ ) ∗ , x ⋆ ∈ C (405) id est, the negative gradient (3.1.8) belongs to the normal cone at x ⋆ as in Figure 59. 2.13.10.1.1 Example. Normal cone to orthant. Consider proper cone K = R n + , the self-dual nonnegative orthant in R n . The normal cone to R n + at a∈ K is (1968) K ⊥ R n + (a∈ Rn +) = −(R n + − a) ∗ = −R n + ∩ a ⊥ , a∈ R n + (406) where −R n += −K ∗ is the algebraic complement of R n + , and a ⊥ is the orthogonal complement of point a . This means: When point a is interior to R n + , the normal cone is the origin. If n p represents the number of nonzero entries in point a∈∂R n + , then dim(−R n + ∩ a ⊥ )= n − n p and there is a complementary relationship between the nonzero entries in point a and the nonzero entries in any vector x∈−R n + ∩ a ⊥ .
2.13. DUAL CONE & GENERALIZED INEQUALITY 183 α α ≥ β ≥ γ β C x ⋆ −∇f(x ⋆ ) γ {z | f(z) = α} {y | ∇f(x ⋆ ) T (y − x ⋆ ) = 0, f(x ⋆ )=γ} Figure 59: Shown is a plausible contour plot in R 2 of some arbitrary differentiable 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’s domain. Function is minimized over convex set C at point x ⋆ iff negative gradient −∇f(x ⋆ ) belongs to normal cone to C there. In circumstance depicted, normal cone is a ray whose direction is coincident with negative gradient. From results in3.1.9 (p.229), ∇f(x ⋆ ) is normal to the γ-sublevel set by Definition E.9.1.0.1.
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2.13. DUAL CONE & GENERALIZED INEQUALITY 183<br />
α<br />
α ≥ β ≥ γ<br />
β<br />
C<br />
x ⋆ −∇f(x ⋆ )<br />
γ<br />
{z | f(z) = α}<br />
{y | ∇f(x ⋆ ) T (y − x ⋆ ) = 0, f(x ⋆ )=γ}<br />
Figure 59: Shown is a plausible contour plot in R 2 of some arbitrary<br />
differentiable real convex function f(x) at selected levels α , β , and γ ;<br />
id est, contours of equal level f (level sets) drawn (dashed) in function’s<br />
domain. Function is minimized over convex set C at point x ⋆ iff negative<br />
gradient −∇f(x ⋆ ) belongs to normal cone to C there. In circumstance<br />
depicted, normal cone is a ray whose direction is coincident with negative<br />
gradient. From results in3.1.9 (p.229), ∇f(x ⋆ ) is normal to the γ-sublevel<br />
set by Definition E.9.1.0.1.
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