# The Linear Programming Problem

• Dimitres Alevras
Part of the Universitext book series (UTX)

## Abstract

Linear programming is the problem of optimizing a linear function subject to finitely many linear constraints in finitely many variables. The standard form of the linear programming problem is
$$\min \left\{ {cx:Ax = b,x \geqslant 0} \right\}$$
for data c ε ℝn, Amxn and b ε ℝm satisfying that the rank of A equals its row rank, i.e., r(A) = m, whereas the canonical form of a linear program is
$$\max \left\{ {cx:Ax \leqslant b,x \geqslant 0} \right\}$$
with (possibly different) data c $$\in \mathbb{R}^{n'} ,A \in \mathbb{R}^{m' \times n'} and b \in \mathbb{R}^{m'}$$. Replacing equations by two inequalities and “free” variables, i.e., variables x j not restricted in sign, by the difference of two nonnegative variables x j = x j + - x j - where x j + ≥ 0 and x j - ≥ 0, by adding slack and/or surplus variables, etc ., any linear program can be brought into either the standard or the canonical form. E.g. by bringing the linear programming problem first into canonical form and then adding slack variables, one obtains a linear program in standard form , t.e., one can indeed assume WROG (=without restriction of generality) that the rank of the constraint matrix A of the linear program in standard form equals its row rank. Apart from the first exercise, which is a pure drilling problem and the answers to which should be obvious to anyone having read the material of Chapter 2 of the book, the other exercises of this chapter are meant to elucidate the importance of inequalities and the usefulness of linear programming in data analysis.

### Keywords

Transportation Drilling