Revisiting Degeneracy, Strict Feasibility, Stability, in Linear Programming

TL;DR

This paper explores how the lack of strict feasibility in linear programming causes degeneracy issues, affecting simplex and interior point methods, and proposes a facial reduction-based preprocessing technique to address these challenges.

Contribution

It establishes a link between strict feasibility and degeneracy in LPs, and introduces an efficient preprocessing method based on facial reduction to improve solution stability.

Findings

Lack of strict feasibility implies all BFS are degenerate.

Existence of a nondegenerate BFS indicates strict feasibility.

The proposed preprocessing method improves numerical stability in LP solutions.

Abstract

Currently, the simplex method and the interior point method are indisputably the most popular algorithms for solving linear programs, LPs. Unlike general conic programs, LPs with a finite optimal value do not require strict feasibility in order to establish strong duality. Hence strict feasibility is seldom a concern, even though strict feasibility is equivalent to stability and a compact dual optimal set. This lack of concern is also true for other types of degeneracy of basic feasible solutions in LP. In this paper we discuss that the specific degeneracy that arises from lack of strict feasibility necessarily causes difficulties in both simplex and interior point methods. In particular, we show that the lack of strict feasibility implies that every basic feasible solution, BFS, is degenerate; thus conversely, the existence of a nondegenerate BFS implies that strict feasibility (regularity) holds. We prove the results using facial reduction and simple linear algebra. In particular, the facially reduced system reveals the implicit non-surjectivity of the linear map of the equality constraint system. As a consequence, we emphasize that facial reduction involves two steps where, the first guarantees strict feasibility, and the second recovers full row rank of the constraint matrix. This illustrates the implicit singularity of problems where strict feasibility fails, and also helps in obtaining new efficient techniques for preproccessing. We include an efficient preprocessing method that can be performed as an extension of phase-I of the two-phase simplex method. We show that this can be used to avoid the loss of precision for many well known problem sets in the literature, e.g., the NETLIB problem set.