An Improved Primal Simplex Algorithm for Degenerate Linear Programs

Abstract

Since its appearance in 1947, the primal simplex algorithm has been one of the most popular algorithms for solving linear programs. It is often very efficient when there is very little degeneracy, but it often struggles in the presence of high degeneracy, executing many pivots without improving the objective function value. In this paper, we propose an improved primal simplex algorithm that deals with this issue. This algorithm is based on new theoretical results that shed light on how to reduce the negative impact of degeneracy. In particular, we show that, from a nonoptimal basic solution with p positive-valued variables, there exists a sequence of at most m - p + 1 simplex pivots that guarantee the improvement of the objective value, where m is the number of constraints in the linear program. These pivots can be identified by solving an auxiliary linear program. Finally, we briefly summarize computational results that show the effectiveness of the proposed algorithm on degenerate linear programs.

Integral Simplex Using Decomposition for the Set Partitioning Problem

“…Proposition 2 (Elhallaoui et al 2011). If CP is infeasible or Z CP 0 then x * C 0 is an optimal solution to .…”

“…We present the results of Elhallaoui et al (2011) using the notation introduced by Padberg (1972, 1975), and Raymond et al (2010). We discuss these results for , the linear relaxation of the set partitioning problem, although they were developed for a general linear programming problem.…”

“…If CP is infeasible or Z CP 0 then x * C 0 is an optimal solution to . Lemma 1 (Elhallaoui et al 2011). If CP is feasible, v * is an optimal solution with Z CP < 0, and S is the subset of indices for which v * j > 0, then adding the variables in S to RP reduces the cost of the solution.…”

“…Proposition 3 (Elhallaoui et al 2011). The convex combination w = j∈S A j v * j is compatible with RP and the set S is minimal in the sense that no convex combination of a strict subset of S is compatible with RP .…”

“…This article also uses results on the improved primal simplex (IPS) algorithm for degenerate linear programs (Elhallaoui et al 2011, Raymond et al 2010. These results are presented in more detail in §2.2.…”