09. NLP (OR Models) - Lecture 9 Nonlinear Programming...

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Unformatted text preview: Lecture 9 Nonlinear Programming Models Topics Convex sets and convex programming First-order optimality conditions Examples Problem classes Minimize f ( x ) s.t. g i ( x ) ( , , =) b i , i = 1,, m x = ( x 1 ,, x n ) T is the n-dimensional vector of decision variables f ( x ) is the objective function g i ( x ) are the constraint functions b i are fixed known constants General NLP Convex Sets Definition : A set S n is convex if every point on the line segment connecting any two points x 1 , x 2 S is also in S . Mathematically, this is equivalent to x = x 1 + (1 ) x 2 S for all such 0 1. x 1 x 2 x 1 x 1 x 2 x 2 x 1 x 2 S = {( x 1 , x 2 ) : (0.5 x 1 0.6) x 2 1 2( x 1 ) 2 + 3( x 2 ) 2 27; x 1 , x 2 0} (Nonconvex) Feasible Region Convex Sets and Optimization Let S = { x n : g i ( x ) b i , i = 1,, m } Fact : If g i ( x ) is a convex function for each i = 1,, m then S is a convex set. Convex Programming Theorem : Let x n and let f ( x ) be a convex function defined over a convex constraint set S . If a finite solution exists to the problem Minimize{ f ( x ) : x S } then all local optima are global optima. If f ( x ) is strictly convex, the optimum is unique. Max f ( x 1 ,, x n ) s.t. g i ( x 1 ,, x n ) b i i = 1,, m x 1 0,, x n 0 is a convex program if f is concave and each g i is convex . Convex Programming Min f ( x 1 ,, x n ) s.t. g i ( x 1 ,, x n ) b i i = 1,, m x 1 0,, x n 0 is a convex program if f is convex and each g i is convex . x 1 1 2 3 4 5 1 2 3 4 5 x 2 Maximize f ( x ) = ( x 1 2) 2 + ( x 2 2) 2 subject to 3 x 1 2 x 2 6 x 1 + x 2 3 x 1 + x 2 7 2 x 1 3 x 2 4 Linearly Constrained Convex Function with Unique Global Maximum (Nonconvex) Optimization Problem First-Order Optimality Conditions Minimize { f ( x ) : g i ( x ) b i , i = 1,, m } Lagrangian: ( 29 1 ( , ) ( ) ( ) m i i i i L f g b = = +- x x x 1 ( , ) ( ) ( ) m i i i L f g = = + = x x x Optimality conditions Stationarity: Complementarity: i g i ( x ) = 0, i = 1,, m Feasibility: g i ( x ) b i , i = 1,, m Nonnegativity: i 0, i = 1,, m Commercial optimization software...
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This note was uploaded on 12/19/2011 for the course M E 366l taught by Professor Staff during the Spring '08 term at University of Texas at Austin.

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09. NLP (OR Models) - Lecture 9 Nonlinear Programming...

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