# Branch-and-cut for linear programs with overlapping SOS1 constraints

@article{Fischer2018BranchandcutFL, title={Branch-and-cut for linear programs with overlapping SOS1 constraints}, author={Tobias Fischer and Marc E. Pfetsch}, journal={Mathematical Programming Computation}, year={2018}, volume={10}, pages={33-68} }

SOS1 constraints require that at most one of a given set of variables is nonzero. In this article, we investigate a branch-and-cut algorithm to solve linear programs with SOS1 constraints. We focus on the case in which the SOS1 constraints overlap. The corresponding conflict graph can algorithmically be exploited, for instance, for improved branching rules, preprocessing, primal heuristics, and cutting planes. In an extensive computational study, we evaluate the components of our implementation…

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## References

SHOWING 1-10 OF 54 REFERENCES

### Branch-and-cut for complementarity-constrained optimization

- Computer ScienceMath. Program. Comput.
- 2014

The results on the use of complementarity cuts within a major commercial optimization solver show that they are of critical importance to tackling difficult CCOP instances, typically reducing the computational time required to solve them tremendously.

### On the structure of linear programs with overlapping cardinality constraints

- Computer ScienceDiscret. Appl. Math.
- 2020

### New Branch-and-Cut Algorithm for Bilevel Linear Programming

- Computer Science
- 2004

The proposed algorithm outperforms pure branch-and-bound when tested on a series of randomly generated problems and also proposes also a set of algorithmic tests and procedures to improve the method.

### Branch-and-cut for combinatorial optimization problems without auxiliary binary variables

- MathematicsThe Knowledge Engineering Review
- 2001

This work presents a branch-and-cut approach that considers the combinatorial constraints without the introduction of binary variables and shows how strong constraints can be derived using ideas from polyhedral combinatorics.

### A Complementarity-based Partitioning and Disjunctive Cut Algorithm for Mathematical Programming Problems with Equilibrium Constraints

- Computer ScienceJ. Glob. Optim.
- 2006

In this paper a branch-and-bound algorithm is proposed for finding a global minimum to a Mathematical Programming Problem with Complementarity (or Equilibrium) Constraints (MPECs), which incorporates…

### On linear programs with linear complementarity constraints

- Computer ScienceJ. Glob. Optim.
- 2012

Several approaches for the global resolution of the LPCC are described, including a logical Benders approach that can be applied to problems that may be infeasible or unbounded.

### On the Global Solution of Linear Programs with Linear Complementarity Constraints

- Computer ScienceSIAM J. Optim.
- 2008

A parameter-free integer-programming-based algorithm for the global resolution of a linear program with linear complementarity constraints (LPCCs), which establishes that the algorithm can handle infeasible, unbounded, and solvable LPCCs effectively.

### Valid inequalities for a single constrained 0-1 MIP set intersected with a conflict graph

- MathematicsDiscret. Optim.
- 2016

### Heuristics for convex mixed integer nonlinear programs

- BusinessComput. Optim. Appl.
- 2012

Diving heuristics, the Feasibility Pump, and Relaxation Induced Neighborhood Search can be adapted in the context of mixed integer nonlinear programming to help find feasible solutions faster and reduce the total solution time of the branch-and-bound algorithm.

### A polyhedral study of the cardinality constrained knapsack problem

- MathematicsMath. Program.
- 2003

Computational results are reported that demonstrate the effectiveness of lifted cover inequalities and the superiority of the approach of not introducing auxiliary 0-1 variables over the traditional MIP approach for this class of problems.