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Unformatted text preview: Atsushi Inoue ECG 765: Mathematical Methods for Economics Fall 2009 Lecture Notes 11 Concave and Convex Functions So far we have focused on finding local optima using firstorder and second order conditions. Local optima are not necessarily global optima in general. Under what conditions are local optima global ones? When the objective function is concave [convex], a local maximum [local minimum] is a global maximum [global minimum]. We will review various characterizations of cocave and convex functions. Outline A. Concave and Convex Functions B. Differentiable Concave and Convex Functions C. TwiceDifferentiable Concave and Convex Functions D. QuasiConcave and QuasiConvex Functions A. Concave and Convex Functions Definition. A set X ⊂ < n is convex if, for every x,y ∈ X and every λ such that 0 ≤ λ ≤ 1, λx + (1 λ ) y ∈ X. Definition. Let f : X → < where X ⊂ < n is convex. f is said to be concave if f ( λx + (1 λ ) y ) ≥ λf ( x ) + (1 λ ) f ( y ) for all x,y ∈ X and all λ such that 0 ≤ λ ≤ 1. f is said to be convex if f ( λx + (1 λ ) y ) ≤ λf ( x ) + (1 λ ) f ( y ) 1 for all x,y ∈ X and all λ such that 0 ≤ λ ≤ 1. Definition. Let f : X → < where X ⊂ < n is convex. f is said to be strictly concave if f ( λx + (1 λ ) y ) > λf ( x ) + (1 λ ) f ( y ) for all x,y ∈ X such that x 6 = y and all λ such that 0 < λ < 1. f is said to be strictly convex if f ( λx + (1 λ ) y ) < λf ( x ) + (1...
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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.
 Fall '07
 FACKLER

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