Improved Primal Simplex: A More General Theoretical Framework and an Extended Experimental Analysis

Omer, Jérémy; Rosat, Samuel; Raymond, Vincent; Soumis, François

doi:10.1287/ijoc.2015.0656

Cited by 8 publications

(21 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The validity of this decomposition is justified by the following theorem of Omer et al [17], which extends to general LPs a result proved by Rosat et al [25] for set partitioning problems. M A N U S C R I P T Theorem 1.…”

Section: Definition 3 (Compatible Variable) a Variable X J Is Compatmentioning

confidence: 89%

“…We then show how good basic solutions can be built to warm-start both the reduced and the complementary problems. The practical impact of these modifications is studied on a large benchmark including the VCS, FA, and UFL instances used in [17] and fourty-five Mittelmann instances. The purpose is to evaluate our new algorithm by comparing it with IPS and the primal simplex of CPLEX, and to show that it is possible to identify quickly the instances that offer a strong potential for faster solution.…”

Section: Contribution Statementmentioning

confidence: 99%

“…Although efficient implementations of linear programming algorithms should focus on LPs with bounded variables, we consider an LP in standard form to clarify and shorten the presentation. Omer et al [17] show the generalization to an LP with bounded variables, and the implementations tested in Section 4 use this generalization.…”

Section: The Improved Primal Simplexmentioning

confidence: 99%

“…To take advantage of degeneracy, Omer et al [17] decompose Gni by referring to the following concept of compatibility.…”

Section: Propositionmentioning

confidence: 99%

See 3 more Smart Citations

A linear programming decomposition focusing on the span of the nondegenerate columns

Omer

Soumis

2015

European Journal of Operational Research

Self Cite

View full text Add to dashboard Cite

To cite this version:Jérémy Omer, François Soumis. A linear programming decomposition focusing on the span of the nondegenerate columns. Highlights• We identify the limits of the existing Improved Primal Simplex (IPS).• We revise every step of the dynamic reduction implemented in IPS.• We show how basic solutions can be built to warm start the solution of each problem.• A simple and fast procedure can test the potential for improvement of the algorithm.• The algorithm outperforms IPS and CPLEXs primal simplex on a large benchmark. AbstractThe improved primal simplex (IPS) was recently developed by Elhalaloui et al. to take advantage of degeneracy when solving linear programs with the primal simplex. It implements a dynamic constraint reduction based on the compatible columns, i.e., those that belong to the span of a given subset of basic columns including the nondegenerate ones. The identification of the compatible variables may however may be computationally costly and a large number of linear programs are solved to enlarge the subset of basic variables. In this article, we first show how the positive edge criterion of Raymond et al. can be included in IPS for a fast identification of the compatible variables. Our algorithm then proceeds through a series of reduction and augmentation phases until optimality is reached. In a reduction phase, we identify compatible variables and focus on them to make quick progress toward optimality. During an augmentation phase, we compute one greatest normalized improving direction and select a subset of variables that should be considered in the reduced problem. Compared with IPS, the linear program that is solved to find this direction involves the data of the original constraint matrix. This new algorithm is tested over Mittelmann's benchmark for linear programming and on instances arising from industrial applications. The results show that the new algorithm outperforms the primal simplex of CPLEX on most highly degenerate instances in which a sufficient number of nonbasic variables are compatible. In contrast, IPS has difficulties on the eleven largest Mittelmann instances. Dealing with degeneracy in the primal simplexStarting from a basic feasible solution, the primal simplex algorithm (see [4]) monotonically improves the objective value by going through a sequence of neighboring feasible bases until optimality is reached. One theoretical limitation of the algorithm is that an iteration may not lead to any progress in the objective value if the solution is degenerate. Geometrically, a degenerate vertex of the n-dimensional feasible polytope of an LP is the intersection of more than n constraints of this LP. In terms of the simplex algorithm, this means that a single vertex can correspond to several bases. The difficulty is that sometimes many iterations move from one basis to another associated with the same vertex. As a consequence, the theoretical convergence of the simplex cannot be guaranteed without a pivoting rule such as those described in [2,3].Although cycling is r...

show abstract

Section: Definition 3 (Compatible Variable) a Variable X J Is Compatmentioning

confidence: 89%

Section: Contribution Statementmentioning

confidence: 99%

Section: The Improved Primal Simplexmentioning

confidence: 99%

“…To take advantage of degeneracy, Omer et al [17] decompose Gni by referring to the following concept of compatibility.…”

Section: Propositionmentioning

confidence: 99%

See 2 more Smart Citations

A linear programming decomposition focusing on the span of the nondegenerate columns

Omer

Soumis

2015

European Journal of Operational Research

Self Cite

View full text Add to dashboard Cite

show abstract

“…For a more recent analysis of the resolution of the pricing problem, Omer et al (2014) explore ways to warm start the basis notably with the use of more elaborate coefficients for the convexity constraints. Alternatively, specialized algorithms can be used in some applications.…”

Section: Algorithmic Stepsmentioning

confidence: 99%

Tools for primal degenerate linear programs: IPS, DCA, and PE

Gauthier

Desrosiers

Lübbecke

2016

EURO Journal on Transportation and Logistics

View full text Add to dashboard Cite

The positive edge pricing rule for the dual simplex

Omer

Towhidi

Soumis

2015

Computers & Operations Research

Self Cite

View full text Add to dashboard Cite

In this article, we develop the two-dimensional positive edge criterion for the dual simplex. This work extends a similar pricing rule implemented by Towhidi et al. [24] to reduce the negative effects of degeneracy in the primal simplex. In the dual simplex, degeneracy occurs when nonbasic variables have a zero reduced cost, and it may lead to pivots that do not improve the objective value. We analyze dual degeneracy to characterize a particular set of dual compatible variables such that if any of them is selected to leave the basis the pivot will be nondegenerate. The dual positive edge rule can be used to modify any pivot selection rule so as to prioritize compatible variables. The expected effect is to reduce the number of pivots during the solution of degenerate problems with the dual simplex. For the experiments, we implement the positive edge rule within the dual simplex of the COIN-OR LP solver, and combine it with both the dual Dantzig and the dual steepest edge criteria. We test our implementation on 62 instances from four well-known benchmarks for linear programming. The results show that the dual positive edge rule significantly improves on the classical pricing rules.

show abstract

Improved Primal Simplex: A More General Theoretical Framework and an Extended Experimental Analysis

Cited by 8 publications

References 19 publications

A linear programming decomposition focusing on the span of the nondegenerate columns

A linear programming decomposition focusing on the span of the nondegenerate columns

Tools for primal degenerate linear programs: IPS, DCA, and PE

The positive edge pricing rule for the dual simplex

Contact Info

Product

Resources

About