# The Dual Simplex Algorithm: Parametric Linear Programming

• Michel Sakarovitch
Part of the Springer Texts in Electrical Engineering book series (STELE)

## Abstract

Consider the pair of dual linear programs
$$(PC)\left\{ \begin{gathered}Ax \leqslant b\;x \geqslant 0 \hfill \\ cx = z(Max) \hfill \\\end{gathered} \right.\quad (DC)\left\{ \begin{gathered} yA \geqslant c\;y \geqslant 0 \hfill \\ yb = w(Min) \hfill \\ \end{gathered} \right.$$
written in canonical form. If b ≥ 0, then x = 0 is a feasible solution of (PC). We say in this case that (PC) is “primal feasible.” If c ≤ 0, then y = 0 is a feasible solution of (DC). In this case, we say that (PC) is “dual feasible.” Given a linear program written in canonical form with respect to a basis, we know (from Theorem IV.3) that this basis is optimal if and only if the linear program is at the same time primal and dual feasible.

## Keywords

Canonical Form Dual Feasibl Parametric Linear Program Piecewise Linear Convex Pivot Operation
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