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09. NLP (OR Models)

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

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Lecture 9 – Nonlinear Programming Models Topics Convex sets and convex programming First-order optimality conditions Examples Problem classes

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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 0 = λ x 1 + (1– λ ) x 2 S for all λ such 0 λ 1. x 1 x 2 x 1 x 1 x 2 x 2

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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.

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Max   f   ( x 1 ,…, x n ) s.t. g i ( x 1 ,…, x n   b i             i  = 1,…, m x  0,…, x  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  0,…, x  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

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(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 μ = = + - x x 1 ( , ) ( ) ( ) m i i i L f g μ = = ∇ + = x x 0 Optimality conditions Stationarity: • Complementarity: μ i g i ( x ) = 0, i = 1,…, m • Feasibility: g i ( x ) b i , i = 1,…, m • Nonnegativity: μ i 0, i = 1,…, m

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Commercial optimization software  cannot  guarantee that a  solution is globally optimal to a nonconvex program.
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09. NLP (OR Models) - Lecture 9 Nonlinear Programming...

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