Linear Programming Problem Solved By a Special Substitution Method

TL;DR

This paper introduces a novel, strongly polynomial special substitution method for solving general linear programming problems, distinct from Fourier-Motzkin elimination, with efficient criteria based on the cost function.

Contribution

The paper develops a unique substitution technique for linear programming that is strongly polynomial and differs from existing elimination methods, maintaining classical backward substitution.

Findings

The method is strongly polynomial.

It does not require matrix inversion.

Backward substitution remains valid for optimal vertex retrieval.

Abstract

In this paper we develop a very special substitution method for solving a general linear programming problem (LPP). Of course the substitution is a kind of elimination of variable but this method must not be confused with the so-called Fourier-Motzkin elimination. The susbtitution developed in this paper only differs by the set of criteria that a variable must verify to be substitued. Most of the criteria are associated with the cost function of the LPP. We prove that the research of the criteria is strongly polynomial. Thus, the special substitution inehrits of the strong polynomiality which characterizes the classical substitution for linear systems. Moreover, as for the classical substitution the backward substitution for finding a vertex associated with the optimum is still valid and does not require to inverse a matrix.