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# Lecture07 - IE 505 MATHEMATICAL PROGRAMMING Bahar Yetis...

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Unformatted text preview: IE 505 MATHEMATICAL PROGRAMMING Bahar Yetis Kara Lecture # 7 Date: 14 October 2007 1 2 3 4 5 6 7 8 LOCAL OPTIMUM Local Minimum Let f: f: X →E where X is a set in R n . A point x Є X is a (strict) local minimum of f over X if there is a neighborhood s.t y } | :| { ) , (      y x y x N X x N y y f x f     ) , ( ) ( ) (  ε 9 Global Minimum A point x Є X is a global minimum of f: X →E over X if: X y y f x f    ) ( ) ( 10 • Theorem: Every local minimum of a convex function f on a convex set X is a global minimum of f over X . • Proof: Exercise 11 OPTIMIZATION • For general optimization problems – Local optimality Global optimality • For LP due to convexity – Local optimality Global optimality 12 LP-OPTIMIZATION • Suppose we are at • And we are questioning moving away from x ’ in the direction of a vector d Є E n . Clearly we should stay in the feasible region....
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Lecture07 - IE 505 MATHEMATICAL PROGRAMMING Bahar Yetis...

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