Lecture 4

X the dual cone of tc is called the normal cone at x

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Unformatted text preview: x). Hence we ¯ ¯ can take kl such that ∥xkl ,l − x∥ < 1/l and ∥yl − αkl ,l (xkl ,l − x)∥ < 1/l. Then ¯ ¯ xkl ,l → x and ¯ ∥y − αkl ,l (xkl ,l − x)∥ ≤ ∥y − yl ∥ + ∥yl − αkl ,l (xkl ,l − x)∥ → 0, ¯ ¯ ¯ ¯ so y ∈ TC (¯). ¯ x The dual cone of TC (¯) is called the normal cone at x and is denoted by x ¯ NC (¯). By the definition of the dual cone, we have x NC (¯) = z ∈ RN (∀y ∈ TC (¯)) ⟨y, z ⟩ ≤ 0 . x x The following theorem is fundamental for constrained optimization. Theorem 2. If f is differentiable and x is a local solution of the problem ¯ minimize f (x) subject to x ∈ C, then −∇f (¯) ∈ NC (¯). x x Proof. By the definition of the normal cone, it suffices to show that ⟨−∇f (¯), y ⟩ ≤ 0 ⇐⇒ ⟨∇f (¯), y ⟩ ≥ 0 x x for all y ∈ TC (¯). Let y ∈ TC (¯) and take a sequence such that αk ≥ 0, xk → x, x x ¯ and αk (xk − x) → y . Since x is a local solution, for sufficiently large k we have ¯ ¯ f (xk ) ≥ f (¯). Since f is differentiable, we have x 0 ≤ f (xk ) − f (¯) = ⟨∇f (¯), xk − x⟩ + o(∥xk − x∥).1 x x ¯ ¯ Multiplying both sides by αk ≥ 0 and letting k → ∞, we get 0 ≤ ⟨∇f (¯), αk (xk − x)⟩ + ∥αk (xk − x)∥ · x ¯ ¯ o(∥xk − x∥) ¯ ∥xk − x∥ ¯ → ⟨∇f (¯), y ⟩ + ∥y ∥ · 0 = ⟨∇f (¯), y ⟩ . x x The geometrical interpretation of Theorem 2 is the following. By the previous lecture, −∇f (¯) is the direction towards which f decreases fastest around x the point x. The tangent cone TC (¯) consists of directions towards which x can ¯ x move around x without violating the constraint x ∈ C . Hence in order for x ¯ ¯ to be a local minimum, −∇f (¯) must make an obtuse angle with any vector in x the tangent cone, for otherwise f can be decreased further. This is the same as −∇f (¯) belonging to the normal cone. x 3 Karush-Kuhn-Tucker theorem Theorem 2 is very general. Usually, we are interested in the cases where the constraint set C is given parametrically. Consider the minimization problem minimize subject to 1 o(h) f (x) gi (x) ≤ 0 hj (x) = 0 (i = 1, . . . , I ) (j = 1, . . . , J ). represents any quantity q (h) such that q (h)/h → 0 as h → 0. (2) 2014W Econ 172B Operations Research (B) Alexis Akira Toda This problem is a special case of problem (1) by setting C = x ∈ RN (∀i)gi (x) ≤ 0, (∀j )hj (x) = 0 . gi (x) ≤ 0 is called an inequality constraint. hj (x) = 0 is an equality constraint. Let x ∈ C be a local solution. To study the shape of C around x, we define...
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This document was uploaded on 02/18/2014 for the course ECON 172b at UCSD.

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