# Optimizing over the split closure

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## Abstract

The polyhedron defined by all the split cuts obtainable directly (i.e. without iterated cut generation) from the LP-relaxation *P* of a mixed integer program (MIP) is termed the (elementary, or rank 1) split closure of *P*. This paper deals with the problem of optimizing over the elementary split closure. This is accomplished by repeatedly solving the following separation problem: given a fractional point, say *x*, find a rank-1 split cut violated by *x* or show that none exists. Following Caprara and Letchford [17], we formulate this separation problem as a nonlinear mixed integer program that can be treated as a parametric mixed integer linear program (PMILP) with a single parameter in the objective function and the right hand side. We develop an algorithmic framework to deal with the resulting PMILP by creating and maintaining a dynamically updated grid of parameter values, and use the corresponding mixed integer programs to generate rank 1 split cuts. Our approach was implemented in the COIN-OR framework using CPLEX 9.0 as a general purpose MIP solver. We report our computational results on well-known benchmark instances from MIPLIB 3.0 and several classes of structured integer and mixed integer problems. Our computational results show that rank-1 split cuts close more than 98% of the duality gap on 15 out of 41 mixed integer instances from MIPLIB 3.0. More than 75% of the duality gap can be closed on an additional 10 instances. The average gap closed over all 41 instances is 72.78%. In the pure integer case, rank-1 split cuts close more than 75% of the duality gap on 13 out of 24 instances from MIPLIB 3.0. On average, rank 1 split cuts close about 72% of the duality gap on these 24 instances. We also report results on several classes of structured problems: capacitated versions of warehouse location, single-source facility location, *p*-median, fixed charge network flow, multi-commodity network design with splittable and unsplittable flows, and lot sizing. The fraction of the integrality gap closed varies for these problem classes between 100 and 67%. We also gathered statistics on the average coefficient size (absolute value) of the disjunctions generated. They turn out to be surprisingly small.

## Keywords

Mixed Integer Balas Master Problem Split Closure Split Disjunction## Preview

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## References

- 1.Ahuja R.K., Orlin J.B., Pallottino S., Scaparra M.P., Scutellà M.G. (2004) A multi-exchange heuristic for the single source capacitated facility location problem. Manage. Sci. 50(6): 749–760CrossRefGoogle Scholar
- 2.Andersen K., Cornuéjols G., Li Y. (2005) Split closure and intersection cuts. Math. Program. A 102, 457–493zbMATHCrossRefGoogle Scholar
- 3.Atamtürk A. (2001) Flow pack facets of the single node fixed-charge flow polytope. Oper. Res. Lett. 29, 107–114zbMATHCrossRefMathSciNetGoogle Scholar
- 4.Atamtürk A., Munóz J.C. (2004) A study of the lot-sizing polytope. Math. Prog. 99, 443–465zbMATHCrossRefGoogle Scholar
- 5.Atamtürk A., Rajan D. (2002) On splittable and unsplittable capacitated network design arc-set polyhedra. Math. Program. 92, 315–333zbMATHCrossRefMathSciNetGoogle Scholar
- 6.Balas E. (1979) Disjunctive programming. Ann. Discrete Math. 5, 3–51zbMATHMathSciNetCrossRefGoogle Scholar
- 7.Balas E., Ceria S., Cornuéjols G. (1993) A lift-and-project cutting plane algorithm for mixed 0–1 programs. Math. Program. 58, 295–324CrossRefGoogle Scholar
- 8.Balas E., Ceria S., Cornuéjols G. (1996) Mixed 0-1 programming by lift-and-project in a branch-and-cut framework. Manage. Sci. 42, 1229–1246zbMATHCrossRefGoogle Scholar
- 9.Balas E., de Souza C. (2005) The vertex separator problem: a polyhedral investigation. Math. Program. 103(3): 583–608zbMATHCrossRefMathSciNetGoogle Scholar
- 10.Balas E., Perregaard M. (2002) Lift and project for mixed 0–1 programming: recent progress. Discrete Appl. Math. 123, 129–154zbMATHCrossRefMathSciNetGoogle Scholar
- 11.Balas E., Perregaard M. (2003) A precise correspondence between lift-and-project cuts, simple disjunctive cuts, and mixed integer Gomory cuts for 0–1 programming. Math. Program. B 94, 221–245zbMATHCrossRefMathSciNetGoogle Scholar
- 12.Balas, E., Saxena, A. Separation functions in disjunctive programming (in preparation)Google Scholar
- 13.Beasley, J.E. OR-Library, people.brunel.ac.uk/∼mastjjb/jeb/info.htmlGoogle Scholar
- 14.Berkeley Computational Optimization Lab, http://ieor.berkeley.edu/~atamturk/dataGoogle Scholar
- 15.Bonami B., Minoux M. (2005) Using rank 1 lift-and-project closures to generate cuts for 0-1 MIP’s, a computational investigation. Discrete optim. 2, 288–308CrossRefMathSciNetGoogle Scholar
- 16.Bonami, P., Cornuéjols, G., Dash, S., Fischetti, M., Lodi, A. Projected Chvátal-Gomory cuts for mixed integer linear programs. Math. Program. A (to appear)Google Scholar
- 17.Caprara A., Letchford A. (2003) On the separation of split cuts and related inequalities. Math. Program. 94, 279–294zbMATHCrossRefMathSciNetGoogle Scholar
- 18.COIN: Computational infrastructure for operations research, http://www.coin-or.orgGoogle Scholar
- 19.Cook W.J., Kannan R., Schrijver A. (1990) Chvatal closures for mixed integer programming problems. Math. Program. 47, 155–174zbMATHCrossRefMathSciNetGoogle Scholar
- 20.Cornuéjols G., Li Y. (2001) Elementary closures for integer programs. Oper. Res. Lett. 28, 1–8zbMATHCrossRefMathSciNetGoogle Scholar
- 21.Dash, S., Günlük, O., Lodi, A. Optimizing over the MIR closure. Talk presented at INFORMS Meeting (San Francisco), November (2005)Google Scholar
- 22.de Souza C., Balas E. (2005) The vertex separator problem: algorithms and computations. Math. Program. 103(3): 609–631zbMATHCrossRefMathSciNetGoogle Scholar
- 23.Fischetti, M., Lodi, A. Optimizing over the first Chvátal closure. Math. Program. B (to appear)Google Scholar
- 24.Holmberg K., Ronnqvist M., Yuan D. (1999) An exact algorithm for the capacited facility location problems with single sourcing. Eur. J. Oper. Res. 113, 544–559zbMATHCrossRefGoogle Scholar
- 25.Nazareth J.L. (1991) The homotopy principle and algorithms for linear programming. SIAM J. Optim. 1, 316–332zbMATHCrossRefMathSciNetGoogle Scholar
- 26.Saxena, A. OSCLIB, http://www.andrew.cmu.edu/user/anureets/osc/osc.htmGoogle Scholar