Coefficient strengthening: a tool for reformulating mixed-integer programs

Providing a good formulation is an important part of solving a mixed-integer program.We suggest measuring the quality of a formulation by whether it is possible to strengthen the coefficients of the formulation. Sequentially strengthening coefficients can then be used as a tool for improving formulations.We believe this method could be useful for analyzing and producing tight formulations of problems that arise in practice.We illustrate the use of the approach on a problem in production scheduling. We also prove that coefficient strengthening leads to formulations with a desirable property: if no coefficient can be strengthened, then no constraint can be replaced by an inequality that dominates it. The effect of coefficient strengthening is tested on a number of problems in computational experiments. The strengthened formulations are compared to reformulations obtained by the preprocessor of a commercial software package. For several test problems, the formulations obtained by coefficient strengthening are substantially stronger than the formulations obtained by the preprocessor. In particular, we use coefficient strengthening to solve two difficult problems to optimality that have only recently been solved.

[1]  Manfred W. Padberg,et al.  Improving LP-Representations of Zero-One Linear Programs for Branch-and-Cut , 1991, INFORMS J. Comput..

[2]  Ellis L. Johnson,et al.  Solving Large-Scale Zero-One Linear Programming Problems , 1983, Oper. Res..

[3]  Gautam Mitra,et al.  Analysis of mathematical programming problems prior to applying the simplex algorithm , 1975, Math. Program..

[4]  Egon Balas.  Disjunctive Programming , 2010, 50 Years of Integer Programming.

[5]  Martin W. P. Savelsbergh,et al.  An Updated Mixed Integer Programming Library: MIPLIB 3.0 , 1998 .

[6]  Yves Pochet,et al.  A tighter continuous time formulation for the cyclic scheduling of a mixed plant , 2008, Comput. Chem. Eng..

[7]  Alexander Martin,et al.  General Mixed Integer Programming: Computational Issues for Branch-and-Cut Algorithms , 2000, Computational Combinatorial Optimization.

[8]  Paolo Toth,et al.  On tightening 0-1 programs based on extensions ofpure 0-1 knapsack and subset-sum problems , 1998, Ann. Oper. Res..

[9]  F. McDonnell.  Inequalities with small coefficients and the reformulation of integer programmes , 1998 .

[10]  Laureano F. Escudero,et al.  Efficient Reformulation for 0-1 Programs - Methods and Computational Results , 1993, Discret. Appl. Math..

[11]  Erling D. Andersen,et al.  Presolving in linear programming , 1995, Math. Program..

[12]  Richard Laundy,et al.  Solving Hard Mixed-Integer Programming Problems with Xpress-MP: A MIPLIB 2003 Case Study , 2009, INFORMS J. Comput..

[13]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .

[14]  Ferydoon Kianfar.  Stronger Inequalities for 0, 1 Integer Programming Using Knapsack Functions , 1971, Oper. Res..

[15]  Dan Wu,et al.  Cyclic short-term scheduling of multiproduct batch plants using continuous-time representation , 2004, Comput. Chem. Eng..

[16]  G. Schilling,et al.  Optimal periodic scheduling of multipurpose plants , 1999 .

[17]  R. Lougee-Heimer,et al.  A Conditional Logic Approach for Strengthening Mixed 0-1 Linear Programs , 2005, Ann. Oper. Res..

[18]  Martin W. P. Savelsbergh,et al.  Preprocessing and Probing Techniques for Mixed Integer Programming Problems , 1994, INFORMS J. Comput..

[19]  Laurence A. Wolsey,et al.  A recursive procedure to generate all cuts for 0–1 mixed integer programs , 1990, Math. Program..

[20]  Egon Balas,et al.  Intersection Cuts - A New Type of Cutting Planes for Integer Programming , 1971, Oper. Res..

[21]  R. Gomory.  AN ALGORITHM FOR THE MIXED INTEGER PROBLEM , 1960 .

[22]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988, Wiley interscience series in discrete mathematics and optimization.

[23]  J. S. Welch,et al.  Finding duplicate rows in a linear programming model , 1986 .

[25]  Laurence A. Wolsey,et al.  Coefficient reduction for inequalities in 0–1 variables , 1974, Math. Program..

[26]  William J. Cook,et al.  Chvátal closures for mixed integer programming problems , 1990, Math. Program..