Zero-one integer programs with few constraints --Lower bounding theory

Abstract The zero-one integer programming problem and its special case, the multiconstraint knapsack problem frequently appear as subproblems in many combinatorial optimization problems. We present several methods for computing lower bounds on the optimal solution of the zero-one integer programming problem. They include Lagrangean, surrogate and composite relaxations. New heuristic procedures are suggested for determining good surrogate multipliers. Based on theoretical results and extensive computational testing, it is shown that for zero-one integer problems with few constraints surrogate relaxation is a viable alternative to the commonly used Lagrangean and linear programming relaxations. These results are used in a follow up paper to develop an efficient branch and bound algorithm for solving zero-one integer programming problems.

[1]  E. Balas,et al.  Pivot and Complement–A Heuristic for 0-1 Programming , 1980 .

[2]  G. Ingargiola,et al.  Reduction Algorithm for Zero-One Single Knapsack Problems , 1973 .

[3]  Egon Balas,et al.  An Algorithm for Large Zero-One Knapsack Problems , 1980, Oper. Res..

[4]  Wei Shih,et al.  A Branch and Bound Method for the Multiconstraint Zero-One Knapsack Problem , 1979 .

[5]  Bezalel Gavish,et al.  An Optimal Solution Method for the Multiple Travelling Salesman Problem , 1980 .

[6]  S. Martello,et al.  An upper bound for the zero-one knapsack problem and a branch and bound algorithm , 1977 .

[7]  E. Balas.  An Additive Algorithm for Solving Linear Programs with Zero-One Variables , 1965 .

[8]  F. Glover.  A Multiphase-Dual Algorithm for the Zero-One Integer Programming Problem , 1965 .

[9]  Ralph E. Gomory,et al.  The Theory and Computation of Knapsack Functions , 1966, Oper. Res..

[10]  Richard Loulou,et al.  New Greedy-Like Heuristics for the Multidimensional 0-1 Knapsack Problem , 1979, Oper. Res..

[11]  Bezalel Gavish,et al.  On obtaining the 'best' multipliers for a lagrangean relaxation for integer programming , 1978, Comput. Oper. Res..

[12]  Les G. Proll,et al.  Algorithms: Algorithm 341: solution of linear programs in 0-1 variables by implicit enumeration , 1968, CACM.

[13]  A Sequential Approach to the $0 - 1$ Linear Programming Problem , 1976 .

[14]  Harvey J. Greenberg,et al.  Surrogate Mathematical Programming , 1970, Oper. Res..

[15]  Harvey J. Greenberg,et al.  The Generalized Penalty-Function/Surrogate Model , 1973, Oper. Res..

[16]  Donald Erlenkotter,et al.  A Dual-Based Procedure for Uncapacitated Facility Location , 1978, Oper. Res..

[17]  Frantisek Fiala.  Computational experience with a modification of an algorithm by Hammer and Rudeanu for 0-1 linear programming , 1971, ACM '71.

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

[19]  Fred Glover,et al.  Surrogate Constraint Duality in Mathematical Programming , 1975, Oper. Res..

[20]  A. M. Geoffrion.  Integer Programming by Implicit Enumeration and Balas’ Method , 1967 .

[21]  David G. Luenberger,et al.  Quasi-Convex Programming , 1968 .

[22]  Philip Wolfe,et al.  Validation of subgradient optimization , 1974, Math. Program..

[23]  Hasan Pirkul,et al.  Zero-one integer programs with few constraints —Efficient branch and bound algorithms , 1985 .

[24]  Boris Polyak.  Minimization of unsmooth functionals , 1969 .

[25]  A. M. Geoffrion,et al.  Lagrangean Relaxation Applied to Capacitated Facility Location Problems , 1978 .

[26]  Bezalel Gavish,et al.  Scheduling and Routing in Transportation and Distribution Systems: Formulations and New Relaxations , 1981 .

[27]  Harvey M. Salkin,et al.  The knapsack problem: A survey , 1975 .

[28]  Harvey J. Everett.  Generalized Lagrange Multiplier Method for Solving Problems of Optimum Allocation of Resources , 1963 .

[29]  Richard M. Karp,et al.  The Traveling-Salesman Problem and Minimum Spanning Trees , 1970, Oper. Res..

[30]  Martin Dyer.  Calculating surrogate constraints , 1980, Math. Program..

[31]  Ronald L. Rardin,et al.  Some relationships between lagrangian and surrogate duality in integer programming , 1979, Math. Program..

[32]  Ellis Horowitz,et al.  Computing Partitions with Applications to the Knapsack Problem , 1974, JACM.

[33]  Robert M. Nauss,et al.  An Efficient Algorithm for the 0-1 Knapsack Problem , 1976 .

[34]  A. Victor Cabot,et al.  An Enumeration Algorithm for Knapsack Problems , 1970, Oper. Res..

[35]  Y. Toyoda.  A Simplified Algorithm for Obtaining Approximate Solutions to Zero-One Programming Problems , 1975 .

[36]  A. L. Soyster,et al.  Zero-one programming with many variables and few constraints , 1978 .