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Unformatted text preview: MthSc 810: Mathematical Programming Lecture 3 Pietro Belotti Dept. of Mathematical Sciences Clemson University September 1, 2011 Reading for today: Sections 1.4, 2.1, 2.2 Reading for Sep. 6: Sections 2.3-2.7 Convex problems Def.: An optimization problem is convex if the objective function is convex all constraints are convex Convex optimization problems are easy : If a problem P is convex, a local optimum x of P is also a global optimum of P . (Hint) When modeling an optimization problem, it would be good if we found a convex problem. Nonconvex problems z o pt = min f ( x ) s . t . f i ( x ) i = 1 , 2 . . . , m What if either of f i , i = , 1 . . . , m is not convex? In order to solve them, We can aim for a feasible solution , whose objective function value is an upper bound z ub z opt. We can obtain a convex relaxation e.g. by eliminating the nonconvex constraints we get an lower bound z lb z opt. Linear Optimization Linear optimization problems are convex . min c x s . t . A x = b x f ( x ) = x and g ( x ) =- x are both convex functions: f ( x + ( 1- ) x ) = x + ( 1- ) x = f ( x ) + ( 1- ) f ( x ) g ( x + ( 1- ) x ) =- ( x + ( 1- ) x ) = g ( x ) + ( 1- ) g ( x ) c f ( x ) and c g ( x ) are convex too for c The objective function is a sum of convex functions...
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This note was uploaded on 03/14/2012 for the course MTHSC 810 taught by Professor Staff during the Fall '08 term at Clemson.
- Fall '08