n11 - Atsushi Inoue ECG 765 Mathematical Methods for...

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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 first-order 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. Twice-Differentiable Concave and Convex Functions D. Quasi-Concave and Quasi-Convex 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
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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 - λ ) f ( y ) for all x, y X such that x 6 = y and all λ
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