Lecture 12

# Lecture 12 - Nonlinear Programming Models and Basic...

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1 Nonlinear Programming − Models and Basic Concepts

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2 Example 1 Minimizing the material used to make a box The volume of the box has to be V Decision: box length: x ; width: y ; height: z ; Objective: To minimize surface area of the box 0 , , subject to 2 2 2 ) , , ( min = + + = z y x V xyz yz xz xy z y x f
3 Example 2 Three warehouses with known locations on a plane described by coordinates ( a i , b i ), i =1, 2, 3 Problem: find a location for a depot so that the total distance to the three warehouses is minimized. Variable: ( x , y ) the coordinates of the depot Model: = - + - = 3 1 2 2 ) ( ) ( ) , ( min i i i b y a x y x f ( a 1 , b 1 ) ( a 2 , b 2 ) ( a 3 , b 3 ) ( x , y )

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4 General NLP x = ( x 1 , x 2 , …, x n ) is an n -dimensional vector f ( x ) = f ( x 1 , x 2 , …, x n ) is the objective function g i ( x ) = g i ( x 1 , x 2 , …, x n ) is a constraint Unconstrained NLP when m = 0 Assumption: – The forms of f ( x ) and g i ( x ) can be arbitrary such as: x 1 x 2 , sin( x 1 ) x 2 , log x 1 , e x , … f ( x ) and g i ( x ) are twice differentiable . ,..., 1 , ) ( subject to ) ( min m i b g f i i = x x
5 Local and Global Optima A local minimum of an NLP is any x in the feasible region such that f ( x ) ≤ f ( x ) for any x in a small neighborhood of x . A global minimum of an NLP is any x 0 in the feasible region such that f ( x 0 )≤ f ( x ) for any x in the entire feasible region Global minimum Local minimum x f ( x )

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6 Local and Global Optima In LP, a local optimum is also a global optimum At least one optimum is at the corner point In NLP, there may exist many local optima that are not global optima A local optimum can be anywhere in the feasible region
7 Convex Function

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Lecture 12 - Nonlinear Programming Models and Basic...

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