# Introduction to Mathematical Programming

• Xiang-Sun Zhang
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 46)

## Abstract

A canonical linear programming problem (CLP) is expressed as follows:
$$(CLP)\;\begin{array}{*{20}{c}} {Minimize\;f(x){\text{ }} = {\text{ }}{{c}^{T}}x} \\ {subject\;to\;Ax = b,\;x0} \\ \end{array}$$
(2.1)
where A = (a ij ) is an m × n matrix (n > m) whose columns are denoted by a i , i = 1,..., n, x = (x 1,...,x n ) T ,c = (c 1,chrw(133),c n ) T ∈ ℝ n , b =(b 1 ,..., b m ) T ∈ ℝ m . A general linear programming problem is
$$(GLP)\begin{array}{*{20}{c}} {Minimize{\text{ }}f(x)} \\ {subject\;to\;{{g}^{x}}0,} \\ {h(x) = 0} \\ \end{array}$$
(2.2)
where f (x), g(x) and h(x)are linear functions. The general LP can be transformed into the canonical form by well-known techniques such as adding slack variables, changing the objective function from positive to negative, splitting a variable without bound into two positively bounded variables, etc. The readers can find these techniques in almost all books about linear programming (to mention a few, [71], [94], [105], [117], [205])

## Keywords

Linear Programming Problem Simplex Method Saddle Point Problem Basic Feasible Solution Integer Linear Programming Problem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media Dordrecht 2000

## Authors and Affiliations

• Xiang-Sun Zhang
• 1
1. 1.Academy of Mathematics and Systems, Institute of Applied MathematicsChinese Academy of SciencesChina