Mixed-integer bilinear programming problems

This paper addresses a class of problems called mixed-integer bilinear programming problems. These problems are identical to the well known bilinear programming problems with the exception that one set of variables is restricted to be binary valued, and they arise in various production, location—allocation, and distribution application contexts. We first identify some special cases of this problem which are relatively more readily solvable, even though their continuous relaxations are still nonconvex. For the more general case, we employ a linearization technique and design a composite Lagrangian relaxation-implicit enumeration-cutting plane algorithm. Extensive computational experience is provided to test the efficacy of various algorithmic strategies and the effects of problem data on the computational effort of the proposed algorithm.

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

[2]  A. M. Geoffrion,et al.  Multicommodity Distribution System Design by Benders Decomposition , 1974 .

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

[4]  Ronald L. Rardin,et al.  Technical Note - Surrogate Constraints and the Strength of Bounds Derived from 0-1 Benders' Partitioning Procedures , 1976, Oper. Res..

[5]  F. Glover.  IMPROVED LINEAR INTEGER PROGRAMMING FORMULATIONS OF NONLINEAR INTEGER PROBLEMS , 1975 .

[6]  Willard I. Zangwill,et al.  Media Selection by Decision Programming , 1976 .

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

[8]  Rafael Lazimy,et al.  Mixed-integer quadratic programming , 1982, Math. Program..

[9]  Fred W. Glover,et al.  Further Reduction of Zero-One Polynomial Programming Problems to Zero-One linear Programming Problems , 1973, Oper. Res..

[10]  Thomas L. Magnanti,et al.  Accelerating Benders Decomposition: Algorithmic Enhancement and Model Selection Criteria , 1981, Oper. Res..

[11]  Warren Philip Adams.  The mixed-integer bilinear programming problem with extensions to zero-one quadratic programs , 1985 .

[12]  Hanif D. Sherali,et al.  On the generation of deep disjunctive cutting planes , 1980 .

[13]  Hanif D. Sherali,et al.  A Decomposition Algorithm for a Discrete Location-Allocation Problem , 1984, Oper. Res..

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

[15]  H. Sherali,et al.  Benders' partitioning scheme applied to a new formulation of the quadratic assignment problem , 1980 .

[16]  Egon Balas,et al.  Duality in Discrete Programming: II. The Quadratic Case , 1969 .

[17]  Hanif D. Sherali,et al.  A versatile scheme for ranking the extreme points of an assignment polytope , 1981 .

[18]  H. Sherali,et al.  On the choice of step size in subgradient optimization , 1981 .

[19]  Harish Vaish,et al.  Nonconvex programming with applications to production and location problems , 1974 .

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

[21]  M. D. Devine,et al.  A Modified Benders' Partitioning Algorithm for Mixed Integer Programming , 1977 .

[22]  A. Victor Cabot,et al.  Solving Certain Nonconvex Quadratic Minimization Problems by Ranking the Extreme Points , 1970, Oper. Res..

[23]  Hiroshi Konno,et al.  A cutting plane algorithm for solving bilinear programs , 1976, Math. Program..

[24]  Harvey M. Salkin,et al.  Integer Programming , 2019, Engineering Optimization Theory and Practice.

[25]  Hanif D. Sherali,et al.  Linearization Strategies for a Class of Zero-One Mixed Integer Programming Problems , 1990, Oper. Res..

[26]  Fred W. Glover,et al.  Technical Note - Converting the 0-1 Polynomial Programming Problem to a 0-1 Linear Program , 1974, Oper. Res..

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

[28]  H. P. Williams.  Experiments in the formulation of integer programming problems , 1974 .

[29]  Lawrence J. Watters.  Letter to the Editor - Reduction of Integer Polynomial Programming Problems to Zero-One Linear Programming Problems , 1967, Oper. Res..

[30]  A. M. Geoffrion.  Generalized Benders decomposition , 1972 .

[31]  Richard E. Wendell,et al.  Minimization of a Non-Separable Objective Function Subject to Disjoint Constraints , 1976, Oper. Res..