Lecture4 - Advanced Mathematical Programming IE417 Lecture...

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Unformatted text preview: Advanced Mathematical Programming IE417 Lecture 4 Dr. Ted Ralphs IE417 Lecture 4 1 Reading for This Lecture • Chapter 3, Sections 1-3 IE417 Lecture 4 2 Convex Functions IE417 Lecture 4 3 Convex Functions Definition 1. Let S be a nonempty convex set on R n . Then the function f : S → R is said to be convex on S if f ( λx 1 + (1- λ ) x 2 ) ≤ λf ( x 1 ) + (1- λ ) f ( x 2 ) for each x 1 ,x 2 ∈ S and λ ∈ (0 , 1) . • Strictly convex means the inequality is strict. • (Strictly) concave is defined analogously. • Can a function be concave and convex? IE417 Lecture 4 4 Properties of Convex Functions • A positive combination of convex functions is convex. • Suppose g : R n → R is concave and let S = { x : g ( x ) > } . Then if f : S → R is defined by f ( x ) = 1 /g ( x ) ,f is convex. • f : S → R is concave if and only if- f is convex. • Let S be a nonempty convex set on R n and let f : S → R be convex. Then the level set S α = { x ∈ S : f ( x ) ≤ α } , where α ∈ R , is a convex set. IE417 Lecture 4 5 Continuity of Convex Functions Theorem 1. Let S be a nonempty convex set on R n and let f : S → R be convex. Then f is continuous on the interior of S . Proof Idea : IE417 Lecture 4 6 Directional Derivative Definition 2. Let S be a nonempty set on R n and let f : S → R be convex. For x * ∈ S and d ∈ R n such that x * + λd ∈ S for λ > sufficiently small. Thesmall....
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This note was uploaded on 08/06/2008 for the course IE 417 taught by Professor Linderoth during the Fall '08 term at Lehigh University .

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Lecture4 - Advanced Mathematical Programming IE417 Lecture...

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