Implementations of special ordered sets in MP software

Abstract Special ordered sets (SOS) have been introduced as a practical device for efficiently handling special classes of nonconvex optimization problems. They are now implemented in most commercial codes for mathematical programming (MP software). The paper gives a survey of possible applications as multiple choice restrictions, conditional multiple choice restrictions, discrete variables, discontinuous variables and piecewise linear functions, global optimization of separable programming problems, alternative right-hand sides, overlapping special ordered sets and the solution of quadratic programming problems. Alternative problem formulations are discussed. Since special ordered sets are not defined uniquely modelling facilities depend on the definition of a special orderedset in a code. The paper demonstrates the superiority of SOS to the application of binary variables if they are treated judiciously.

[1]  A. Land,et al.  Computer Codes for Problems of Integer Programming , 1979 .

[2]  Dennis J. Sweeney,et al.  An ideal column algorithm for integer programs with special ordered sets of variables , 1983, Math. Program..

[3]  G. Dantzig.  ON THE SIGNIFICANCE OF SOLVING LINEAR PROGRAMMING PROBLEMS WITH SOME INTEGER VARIABLES , 1960 .

[4]  E. M. L. Beale,et al.  Global optimization using special ordered sets , 1976, Math. Program..

[5]  Jean-Michel Gauthier,et al.  Experiments in mixed-integer linear programming using pseudo-costs , 1977, Math. Program..

[6]  E. M. L. Beale,et al.  Mathematical programming in practice , 1968 .

[7]  E.M.L. Beale,et al.  Branch and Bound Methods for Mathematical Programming Systems , 1977 .

[8]  Prabhakant Sinha,et al.  Integer Programming Models for Sales Resource Allocation , 1980 .

[9]  Prabhakant Sinha,et al.  An Allocation Model for Catalog Space Planning , 1979 .

[10]  George B. Dantzig,et al.  Generalized Upper Bounding Techniques , 1967, J. Comput. Syst. Sci..

[11]  Michel Bénichou,et al.  The efficient solution of large-scale linear programming problems—some algorithmic techniques and computational results , 1977, Math. Program..

[12]  R. J. Dakin,et al.  A tree-search algorithm for mixed integer programming problems , 1965, Comput. J..

[13]  P. Wolfe.  THE SIMPLEX METHOD FOR QUADRATIC PROGRAMMING , 1959 .

[14]  W. C. Healy.  Multiple Choice Programming A Procedure for Linear Programming with Zero-One Variables , 1964 .