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Lecture15

Lecture15 - C&O 355 Mathematical Programming Fall 2010...

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C&O 355 Mathematical Programming Fall 2010 Lecture 15 N. Harvey

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Topics Minimizing over a convex set: Necessary & Sufficient Conditions (Mini)- KKT Theorem Minimizing over a polyhedral set: Necessary & Sufficient Conditions Smallest Enclosing Ball Problem
Proof: ( direction Direct from subgradient inequality. (Theorem 3.5) f(z) ¸ f(x) + r f(x) T (z-x) ¸ f(x) Subgradient inequality Our hypothesis Thm 3.12: Let C µ R n be a convex set. Let f : R n ! R be convex and differentiable. Then x minimizes f over C iff r f(x) T (z-x) ¸ 0 8 z 2 C. Minimizing over a Convex Set: Necessary & Sufficient Conditions

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Minimizing over a Convex Set: Necessary & Sufficient Conditions Thm 3.12: Let C µ R n be a convex set. Let f : R n ! R be convex and differentiable. Then x minimizes f over C iff r f(x) T (z-x) ¸ 0 8 z 2 C. Proof: ) direction Let x be a minimizer, let z 2 C and let y = z-x. Recall that r f(x) T y = f’(x;y) = lim t ! 0 f(x+ty)-f(x). If limit is negative then we have f(x+ty)<f(x) for some t 2 [0,1], contradicting that x is a minimizer. So the limit is non-negative, and r f(x) T y ¸ 0. ¥ t
Smallest Ball Problem Let { p 1 ,…, p n } be points in R d . Find (unique!) ball (not an ellipsoid!) of smallest volume that contains all the p i ’s. In other words, we want to solve: min { r : 9 y 2 R d s.t. p i 2 B( y , r ) 8 i } y r

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Smallest Ball Problem Let { p 1 ,…, p n } be points in R d . Find (unique!) ball (not an ellipsoid!) of smallest volume that contains all the p i ’s. In other words, we want to solve: min { r : 9 y 2 R d s.t. p i 2 B( y , r ) 8 i } We will formulate this as a convex program. In fact, our convex program will be of the form min { f(x) : Ax=b, x ¸ 0 }, where f is convex.
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