Lecture5

# Lecture5 - Advanced Mathematical Programming IE417 Lecture...

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Advanced Mathematical Programming IE417 Lecture 5 Dr. Ted Ralphs

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IE417 Lecture 5 1 Reading for This Lecture Chapter 3, Sections 4-5 Chapter 4, Section 1
IE417 Lecture 5 2 Maxima and Minima of Convex Functions

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IE417 Lecture 5 3 Minimizing a Convex Function Theorem 1. Let S be a nonempty convex set on R n and let f : S R be convex on S . Suppose that x * is a local optimal solution to min x S f ( x ) . Then x * is a global optimal solution. If either x * is a strict local optimum or f is strictly convex, then x * is the unique global optimal solution.
IE417 Lecture 5 4 Necessary and Suﬃcient Conditions Theorem 2. Let S be a nonempty convex set on R n and let f : S R be convex on S . The point x * S is an optimal solution to the problem min x S f ( x ) if and only if f has a subgradient ξ such that ξ T ( x - x * ) 0 x S . This implies that if S is open, then x * is an optimal solution if and only if there is a zero subgradient of f at x * .

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IE417 Lecture 5 5 Alternative Optima Theorem 3. Let
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Lecture5 - Advanced Mathematical Programming IE417 Lecture...

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