Exact solutions to linear programming problems

Applegate, David; Cook, William J.; Dash, Sanjeeb; Espinoza, Daniel

doi:10.1016/j.orl.2006.12.010

Cited by 94 publications

(144 citation statements)

References 12 publications

(14 reference statements)

Supporting

Mentioning

139

Contrasting

Unclassified

Order By: Relevance

“…Table 2 shows the times required to generate and solve the shrunken LPs. Both CPLEX, version 12, and and QSopt ex, a rational LP solver [2], were used to solve the LP problems. CPLEX was used to solve both the formulation with and without counting constraints, while QSopet ex was used to solve only the formulations with the counting constraints.…”

Section: Computational Resultsmentioning

confidence: 99%

Using symmetry to optimize over the Sherali–Adams relaxation

Ostrowski

2014

Math. Prog. Comp.

View full text Add to dashboard Cite

In this paper we examine the impact of using the Sherali-Adams procedure on highly symmetric integer programming problems. Linear relaxations of the extended formulations generated by Sherali-Adams can be very large, containing O( n d ) many variables for the level-d closure. When large amounts of symmetry are present in the problem instance however, the symmetry can be used to generate a much smaller linear program that has an identical objective value. We demonstrate this by computing the bound associated with the level 1, 2, and 3 relaxations of several highly symmetric binary integer programming problems. We also present a class of constraints, called counting constraints, that further improves the bound, and in some cases provides a tight formulation. A major advantage of the Sherali-Adams formulation over the traditional formulation is that symmetry-breaking constraints can be more efficiently implemented. We show that using the Sherali-Adams formulation in conjunction with the symmetry breaking tool isomorphism pruning can lead to the pruning of more nodes early on in the branch-and-bound process.

show abstract

Section: Computational Resultsmentioning

confidence: 99%

Using symmetry to optimize over the Sherali–Adams relaxation

Ostrowski

2014

Math. Prog. Comp.

View full text Add to dashboard Cite

show abstract

“…This is a pertinent issue to the generation of deep cuts and is also the reason commercial solvers refrain from generating many rounds of MIR cuts (Cook et al 2009). To circumvent this problem, Chvátal et al (2013) use the rational solver they developed in Applegate et al (2007). In addition, they provide a floating point implementation of their method to compare their results with other studies.…”

Section: Computational Considerationsmentioning

confidence: 99%

Local Cuts and Two-Period Convex Hull Closures for Big-Bucket Lot-Sizing Problems

Akartunalı

Fragkos

Miller³

et al. 2016

INFORMS Journal on Computing

View full text Add to dashboard Cite

D espite the significant attention they have drawn, big-bucket lot-sizing problems remain notoriously difficult to solve. Previous literature contained results (computational and theoretical) indicating that what makes these problems difficult are the embedded single-machine, single-level, multiperiod submodels. We therefore consider the simplest such submodel, a multi-item, two-period capacitated relaxation. We propose a methodology that can approximate the convex hulls of all such possible relaxations by generating violated valid inequalities. To generate such inequalities, we separate two-period projections of fractional linear programming solutions from the convex hulls of the two-period closure we study. The convex hull representation of the twoperiod closure is generated dynamically using column generation. Contrary to regular column generation, our method is an outer approximation and can therefore be used efficiently in a regular branch-and-bound procedure. We present computational results that illustrate how these two-period models could be effective in solving complicated problems.

show abstract

“…A combination of the rational LP solver [4] with the Branch-and-Cut code SCIP [1] is in progress. The idea is to use finite precision arithmetic for the majority of the computations, and switch to (slower) rational arithmetic only for those operations that would invalidate optimality of the result if carried out in a non-exact fashion (such as pruning based on dual bounds).…”

Section: Related Workmentioning

confidence: 99%

“…The feasibility and integrality tolerances for Cplex were set to 10 −9 , the smallest feasibility tolerance allowed by Cplex 2 . The rational LP solver used by GenerateSolutions is QSopt ex [4].…”

Section: A1 Implementationmentioning

confidence: 99%

On the safety of Gomory cut generators

2013

View full text Add to dashboard Cite

Gomory mixed-integer cuts are one of the key components in Branch-and-Cut solvers for mixed-integer linear programs. The textbook formula for generating these cuts is not used directly in open-source and commercial software that work in finite precision: Additional steps are performed to avoid the generation of invalid cuts due to the limited numerical precision of the computations. This paper studies the impact of some of these steps on the safety of Gomory mixed-integer cut generators. As the generation of invalid cuts is a relatively rare event, the experimental design for this study is particularly important. We propose an experimental setup that allows statistically significant comparisons of generators. We also propose a parameter optimization algorithm and use it to find a Gomory mixedinteger cut generator that is as safe as a benchmark cut generator from a commercial solver even though it generates many more cuts.

show abstract

Exact solutions to linear programming problems

Cited by 94 publications

References 12 publications

Using symmetry to optimize over the Sherali–Adams relaxation

Using symmetry to optimize over the Sherali–Adams relaxation

Local Cuts and Two-Period Convex Hull Closures for Big-Bucket Lot-Sizing Problems

On the safety of Gomory cut generators

Contact Info

Product

Resources

About