Lecture 2

# Lecture 2 - 540:311 DETERMINISTIC MODELS IN OPERATIONS...

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540:311 DETERMINISTIC MODELS IN OPERATIONS RESEARCH Lecture 2: Chapter 2 – 3.1 Class Meeting: Mon Jan 24 th 10:20-11:40am Recitation: Basic linear algebra (matrix operations) Prof. W. Art Chaovalitwongse

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A Linear Programming Problem is an optimization problem for which we do the following: 1. Attempt to maximize (minimize) a linear function of the decision variables . (called objective function ) 2. Ensure that the values of the decision variables satisfy a set of linear constraints. 3. Guarantee a sign restriction associated with each variable.
Key to Success From the problem statement, IDENTIFY Decision variables: – the quantities that can vary; Objective function: – the expression that is being minimized or maximized; min f(x) = max - f(x) Constraints: – equations and inequalities that the decision variables must satisfy

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Examples of linear expressions 17 x 1 57 x 2 + 91 x 3 or A 3 . 4 B + 2 C. The variables of the first expression are x 1 , x 2 , x 3 , and of the second — A,B, C . Both expressions depend on their respective variables linearly. A linear constraint requires a linear expression to be equal to a number, greater-than-or-equal-to a number, or less-than-or-equal-to a number, as illustrated below: 2 A 3 B = 6 and A 3 . 4 B + 2 C 2 and C 0 . (In this class, as is typical in this field, we will not consider the inequality to be linear constraint!). . Why? 17 x 1 57 x 2 + 91 x 3 > 12
Representing LPs a i x = b i a i x b i and a i x b i • where a i is a notation for the i th row of matrix A • a system of inequalities • a system of equalities • both are equivalent: a i x b i a i x + s = b i • where s 0 is a non-negative slack variable. • Slack variables are important, since they allow us to pass from inequalities to equalities.

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Non-negativity Issues Non-negativity can be enforced by inequalities: x 0, for any vector x Unrestricted variables can be represented as a sum of nonnegative variables: x = x ` – x z where x is an unrestricted variable and x ` , x z are non- negative variables. Thus, the general LP requires that all x 0 LP is also called cone programming in the non-negative cone (the positive orthant is a cone)
Basic Linear Algebra • Basic tool for understanding LP • Main components are (1) matrices; (2) vectors • Basic operations of matrices – Inverse – Transpose • The rank of a matrix is the number of linear independent rows. • If nxn matrix has rank n , the matrix is said to have full rank. Attend recitation class Wednesday Jan 26 th !

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Chapter 2 - Learning Objectives Matrices and vectors with basic matrix operations. Matrices and systems of linear equations. Gauss-Jordan method for solving linear equations. Concepts of – linearly independent set of vectors, – linearly dependent set of vectors, – rank of a matrix.
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Lecture 2 - 540:311 DETERMINISTIC MODELS IN OPERATIONS...

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