Chapter 1 - MA3252 LINEAR AND NETWORK OPTIMIZATION Topic 1:...

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Unformatted text preview: MA3252 LINEAR AND NETWORK OPTIMIZATION Topic 1: Introduction to Linear Programming 1 / 51 Linear programming is a mathematical modeling technique designed to minimize or maximize a linear cost function subject to a finite set of linear equality and inequality constraints. It is the basis for the development of solution algorithms of other (more complex) types of operations research (OR) models, including integer, nonlinear, and stochastic programming. 2 / 51 History of Optimization • Fermat and Newton, 17th century: min f ( x ) x scalar Find x such that df ( x ) dx = 0 3 / 51 • Euler, 18th century: min f ( x 1 ,..., x n ) Find x 1 ,..., x n such that ∇ f ( x 1 ,..., x n ) =           ∂ f ( x 1 ,..., x n ) ∂ x 1 ∂ f ( x 1 ,..., x n ) ∂ x 2 . . . ∂ f ( x 1 ,..., x n ) ∂ x n           = 0 4 / 51 • Lagrange, 18th century: min f ( x 1 ,..., x n ) s . t . g k ( x 1 ,..., x n ) = 0 , k = 1 ,..., m . 5 / 51 A general optimization problem is to find the “best” solution ( x 1 ,..., x n ) from a feasible region P where the “best” refers to the minimum (or maximum) value of f ( x 1 ,..., x n ): min (or max) f ( x 1 ,..., x n ) s . t . ( x 1 ,..., x n ) ∈ P . 6 / 51 General Linear Programming Problem In this section, the general linear programming problem is introduced followed by some examples to help familiarize with some basic terminology used in LP. 7 / 51 Notation 1 For a matrix A , we denote its transpose by A . 2 An n-dimensional vector x ∈ R n is denoted by a column vector: x =      x 1 x 2 . . . x n      . 3 For vectors x = ( x 1 , x 2 , ··· , x n ) and y = ( y 1 , y 2 , ··· , y n ) , the inner product is defined as: x y = y x = n X i =1 x i y i = x 1 y 1 + x 2 y 2 + ··· + x n y n . 8 / 51 In a linear programming problem, a cost (or profit) vector c = ( c 1 , c 2 , ··· , c n ) is given. The objective is to minimize (or maximize) the linear objective function c x over all vectors x = ( x 1 , x 2 , ··· , x n ) , subject to a finite set of linear equality and inequality constraints. This can be summarized as: min (or max) c x s . t . a i x ≥ b i , i ∈ M + , a i x ≤ b i , i ∈ M- , a i x = b i , i ∈ M , x j ≥ , j ∈ N + , x j ≤ , j ∈ N- , where a i = ( a i 1 , a i 2 , a i 3 , ··· , a in ) is a vector in R n and b i is a scalar. 9 / 51 Terminology 1 Variables x i are called decision variables . There are n of them. 2 Each constraint is either an equality or inequality of the form ≤ or ≥ . 3 If j is in neither N + nor N- , there are no restrictions on the sign of x j . The variable x j is then said to be a free or unrestricted variable ....
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This note was uploaded on 12/02/2011 for the course MATH 3252 taught by Professor Tanbanpin during the Spring '10 term at National University of Singapore.

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Chapter 1 - MA3252 LINEAR AND NETWORK OPTIMIZATION Topic 1:...

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