n17 - Atsushi Inoue ECG 765 Mathematical Methods for...

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Unformatted text preview: Atsushi Inoue ECG 765: Mathematical Methods for Economics Term 1, 2009 Lecture Notes 17 Concave Programming The Lagrange and Kuhn-Tucker theorems provide necessary conditions for constraint optima. To obtain sufficient conditions we often need to impose additional conditions on objective and constraint functions. Outline A. Optimization by Separation B. Concave Programming C. Quasi-Concave Programming A. Optimization by Separation Definition. Let p be an n × 1 nonzero vector and let a ∈ < . The set H defined by H = { x ∈ < n : p · x = a } is called a hyperplane in < n and is denoted by H ( p,a ). Separation Theorem 1. Let A be a nonempty convex set in < n and let x * be a point in < n \ A . Then there is a hyperplane H ( p,a ) in < n with p 6 = 0 and a = p · x * which separates X and x * : p · x * ≤ p · x, for all x ∈ A, Separation Theorem 2. Let A and B be convex sets in < n such that A ∩ B = ∅ . Then there exists a hyperplane H ( p,a ) in < n which separates A and B : p · x ≤ a, for all x ∈ A, p · x ≥ a, for all x ∈ B. 1 Theorem. Let f : X → < be quasi-concave and g : X → < be quasi-convex. x * maximizes f ( x ) subject to the constraint g ( x ) ≤ c if and only if there is a nonzero n × 1 vector p such that (i) x * maximizes p · x subject to g ( x ) ≤ c ; and (ii) x * minimizes p · x subject to f ( x ) ≥ f ( x * )....
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This note was uploaded on 10/22/2009 for the course ECG 765 taught by Professor Fackler during the Fall '07 term at N.C. State.

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n17 - Atsushi Inoue ECG 765 Mathematical Methods for...

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