On the existence of a short pivoting sequence for a linear program

TL;DR

This paper proves the theoretical existence of a short pivoting sequence in linear programming that incrementally increases the submatrix size, leading to optimal solutions, without providing a method to generate such sequences.

Contribution

It introduces a novel existence proof for a short pivoting sequence in linear programs, based on basis matrix decomposition, without offering a constructive algorithm.

Findings

Existence of a pivoting sequence bounded by the minimum dimension of the constraint matrix.

Sequence creates nonsingular submatrices increasing by one row and column each step.

Final optimal solutions are obtained after a finite sequence of pivots.

Abstract

Pivoting methods are of vital importance for linear programming, the simplex method being the by far most well-known. In this paper, a primal-dual pair of linear programs in canonical form is considered. We show that there exists a sequence of pivots, whose length is bounded by the minimum dimension of the constraint matrix, such that the pivot creates a nonsingular submatrix of the constraint matrix which increases by one row and one column at each iteration. Solving a pair of linear equations for each of these submatrices generates a sequence of optimal solutions of a primal-dual pair of linear programs of increasing dimensions, originating at the origin. The optimal solutions to the original primal-dual pair of linear programs are obtained in the final step. It is only an existence result, we have not been able to generate any rules based on properties of the problem to generate the sequence. The result is obtained by a decomposition of the final basis matrix.