Just MIP it!
Modern Mixed-Integer Programming (MIP) solvers exploit a rich arsenal of tools to attack hard problems. It is idely accepted by the OR community that the solution of very hard MIPs can take advantage from the solution of a series of time-consuming auxiliary Linear Programs (LPs) intended to enhance the performance of the overall MIP solver. For instance, auxiliary LPs may be solved to generate powerful disjunctive cuts, or to implement a strong branching policy. Also well established is the fact that finding good-quality heuristic MIP solutions often requires a computing time that is just comparable to that needed to solve the LP relaxations. So, it makes sense to think of a new generation of MIP solvers where auxiliary MIPs (as opposed to LPs) are heuristically solved on the fly, with the aim of bringing the MIP technology under the chest of the MIP solver itself. This leads to the idea of “translating into a MIP model” (MIPping) some crucial decisions to be taken within a MIP algorithm (How to cut? How to improve the incumbent solution? Is the current node dominated?). In this paper we survey a number of successful applications of the above approach.
KeywordsHull Ceria Octane Balas DMPA
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This work was supported by the Future and Emerging Technologies unit of the EC (IST priority), under contract no. FP6-021235-2 (project “ARRIVAL”) and by MiUR, Italy.
- 9.R.E. Bixby, S. Ceria, C.M. McZeal, and M.W.P. Savelsbergh. An updated mixed integer programming library: MIPLIB 3.0. Optima, 58:12–15, 1998.Google Scholar
- 14.G. Codato and M. Fischetti. Combinatorial Benders cuts. In D. Bienstock and G. Nemhauser, editors, Integer Programming and Combinatorial Optimization, IPCO X, volume 3064 of Lecture Notes in Computer Science, pages 178–195. Springer, 2004.Google Scholar
- 19.S. Dash, O. Günlük, and A. Lodi. On the MIR closure of polyhedra. In M. Fischetti and D.P. Williamson, editors, Integer Programming and Combinatorial Optimization, IPCO XII, volume 4513 of Lecture Notes in Computer Science, pages 337–351. Springer, 2007.Google Scholar
- 20.S. Dash, O. Günlük, and A. Lodi. MIR closures of polyhedral sets. Mathematical Programming, DOI 10.1007/s10107-008-0225-x, 2008.Google Scholar
- 21.Double-Click sas. personal communication, 2001.Google Scholar
- 31.J. Gleeson and J. Ryan. Identifying minimally infeasible subsystems of inequalities. ORSA Journal on Computing, 2:61–63, 1990.Google Scholar
- 35.F. Glover and M. Laguna. Tabu Search. Kluwer, 1997.Google Scholar
- 36.R.E. Gomory. An algorithm for the mixed integer problem. Technical Report RM-2597, The Rand Corporation, 1960.Google Scholar
- 37.M. Grötschel, L. Lovász, and A. Schrijver. Geometric Algorithms and Combinatorial Optimization. Springer-Verlag, 1988.Google Scholar
- 40.T. Ibaraki, T. Ohashi, and F. Mine. A heuristic algorithm for mixed-integer programming problems. Mathematical Programming Study, 2:115–136, 1974.Google Scholar
- 41.ILOG S.A. CPLEX: ILOG CPLEX 11.0 User’s Manual and Reference Manual, 2007. http://www.ilog.com.
- 42.G.W. Klau. personal communication, 2002.Google Scholar
- 43.A. Løkketangen. Heuristics for 0-1 mixed-integer programming. In P.M. Pardalos and M.G.C. Resende, editors, Handbook of Applied Optimization, pages 474–477. Oxford University Press, 2002.Google Scholar
- 46.S. Martello and P. Toth. Knapsack Problems: Algorithms and Computer Implementations. Wiley, New York, 1990.Google Scholar
- 47.A.J. Miller. personal communication, 2003.Google Scholar
- 51.C.H. Papadimitriou and K. Steiglitz. Combinatorial Optimization: Algorithms and Complexity. Prentice-Hall, 1982.Google Scholar
- 53.E. Rothberg. personal communication, 2002.Google Scholar
- 54.D. Salvagnin. A dominance procedure for integer programming. Master’s thesis, University of Padua, October 2005.Google Scholar