n5 - Atsushi Inoue ECG 765 Mathematical Methods for...

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Unformatted text preview: Atsushi Inoue ECG 765: Mathematical Methods for Economics Fall 2009 Lecture Notes 5 Local and Global Optima We will review basic concepts in optimization and differential calculus for univariate functions. Our goal is to characterize optimal solutions using first and second derivatives of objective functions. Outline A. Local and Global Optima B. Univariate Differentiable Functions A. Local and Global Optima Definition. Let f be a function defined on X . We say that f has a local maximum [ local minimum ] at a point x ∈ X if there exists ε > 0 such that f ( x ) ≥ f ( x ) [ f ( x ) ≤ f ( x )] for all x ∈ B ε ( x ) ∩ X . We say that f has a strict local maximum [ local minimum ] at a point x ∈ X if there exists ε > such that f ( x ) > f ( x ) [ f ( x ) < f ( x )] for all x ∈ B ε ( x ) ∩ X such that x 6 = x . Definition. Let f be a function defined on X . We say that f has a global maximum [ global minimum ] at a point x ∈ X if f ( x ) ≥ f ( x ) [ 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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n5 - Atsushi Inoue ECG 765 Mathematical Methods for...

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