Subset Coefficient Reduction Cuts for 0/1 Mixed-Integer Programming

We describe a method for generating cuts for mixed-integer 0/1 programs. These cuts are designed to tighten an integer program prior to applying linear programming based branch and bound algorithms. The method involves two basic ideas: subset selection and coefficient reduction. Coefficient reduction is a process of reducing the coefficients of the 0/1 variables. Subset selection is combined with coefficient reduction by applying the coefficient reduction process to the coefficients of a subset of variables from the constraints in the problem formulation. The paper exploits these two simple ideas to derive a broad class of cuts for integer programs with both 0/1 and continuous variables. It also reports on the use of this methodology in solving a variety of fixed charge problems.

[1]  Arthur M. Geoffrion,et al.  An Improved Implicit Enumeration Approach for Integer Programming , 1969, Oper. Res..

[2]  Robert G. Jeroslow,et al.  An Introduction to the Theory of Cutting-Planes , 1979 .

[3]  Manfred W. Padberg.  (1,k)-configurations and facets for packing problems , 1980, Math. Program..

[4]  Laurence A. Wolsey,et al.  Technical Note - Facets and Strong Valid Inequalities for Integer Programs , 1976, Oper. Res..

[5]  J. F. Benders.  Partitioning procedures for solving mixed-variables programming problems , 1962 .

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

[7]  Laurence A. Wolsey,et al.  Faces for a linear inequality in 0–1 variables , 1975, Math. Program..

[8]  Ellis L. Johnson,et al.  Some continuous functions related to corner polyhedra , 1972, Math. Program..

[9]  L. A. Oley,et al.  Automatic reformulation of mixed and pure integer models to reduce solution time in apex IV , 1983, SMAP.

[10]  Egon Balas,et al.  Facets of the knapsack polytope , 1975, Math. Program..

[11]  Monique Guignard-Spielberg,et al.  Logical Reduction Methods in Zero-One Programming - Minimal Preferred Variables , 1981, Oper. Res..

[12]  Laurence A. Wolsey,et al.  Valid Linear Inequalities for Fixed Charge Problems , 1985, Oper. Res..

[13]  Manfred W. Padberg,et al.  On the facial structure of set packing polyhedra , 1973, Math. Program..

[14]  M. Padberg.  Covering, Packing and Knapsack Problems , 1979 .

[15]  H. Crowder,et al.  Solving Large-Scale Symmetric Travelling Salesman Problems to Optimality , 1980 .

[16]  E. L. Johnson,et al.  Integer Programming Codes , 1978 .

[17]  M. Padberg,et al.  On the symmetric travelling salesman problem II , 1979 .

[18]  Uwe H. Suhl,et al.  Experiments in integer programming , 1980, Discret. Appl. Math..

[19]  Egon Balas,et al.  Discrete Programming by the Filter Method , 1967, Oper. Res..

[20]  P. Gray,et al.  Solving Fixed Charge Location-Allocation Problems with Capacity and Configuration Constraints , 1971 .

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

[22]  Eitan Zemel.  Lifting the Facets of 0-1 Polytopes. , 1974 .

[23]  E. Balas,et al.  Canonical Cuts on the Unit Hypercube , 1972 .

[24]  Linus E Schrage.  User's manual for LINDO , 1981 .

[25]  Manfred W. Padberg,et al.  On the symmetric travelling salesman problem: A computational study , 1980 .

[26]  G. L. Thompson,et al.  A Heuristic Approach to Solving Travelling Salesman Problems , 1964 .