The Non-Substitution Theorem, Uniqueness of Solution and Convex combinations of basic optimal solutions for linear optimization

TL;DR

This paper presents simplified proofs of fundamental theorems in linear programming, including the non-substitution theorem, uniqueness of solutions, and convex combinations of basic optimal solutions, avoiding advanced polyhedral theory.

Contribution

It provides straightforward proofs of key linear programming theorems, including a new proof of the Birkhoff-von Neumann Theorem, using basic lemmas and elementary arguments.

Findings

Set of optimal solutions is convex and formed by convex combinations of basic solutions.

Basic optimal solutions are the extreme points of the solution set.

Proofs avoid advanced polyhedral and combinatorial results.

Abstract

Our first result is a statement of a somewhat general form of a non-substitution theorem for linear programming problems, along with a very easy proof of the same. Subsequently, we provide an easy proof of theorem 1 in a 1979 paper of Olvi L. Mangasarian, based on a new result in terms of two statements that are each equivalent to a given solution of a linear programming problem being its unique solution. We also provide a simple proof of the result that states that the set of optimal solutions of a bounded linear optimization problem is the set of all convex combinations of its basic optimal solutions and the set of basic optimal solutions are the extreme points of the set of optimal solutions. We do so by appealing to the lemma due to Farkas and the well-known result that states that if a linear optimization problem has an optimal solution, it has at least one basic optimal solution. Both results we appeal to have easy proofs. We do not appeal to any version of the Klein-Milman Theorem or any result in advanced polyhedral combinatorics to obtain our results. As an application of this result, we obtain a simple proof of the Birkhoff-von Neumann Theorem.