MIP: Theory and Practice - Closing the Gap

For many years the principal solution technique used in the practice of mixed-integer programming has remained largely unchanged: Linear programming based branch-and-bound, introduced by Land and Doig (1960). This, in spite of the fact that there has been significant progress in the theory of integer programming and in the closely related field of combinatorial optimization. Many of the ideas developed there have received extensive computational testing, but, until recently, relatively little of that work has made it into the commercial codes used by practitioners. That situation has now changed. Several such codes, among them LINGO1, OSL2, and XPRESS-MP3, as well as the CPLEX4 code studied in this paper, now include cutting-plane capabilities as well as other ideas from the backlog of accumulated theory. As suggested by the title of this paper, the gap between theory and practice is indeed closing.

[1]  Anoop Gupta,et al.  Efficient sparse matrix factorization on high performance workstations—exploiting the memory hierarchy , 1991, TOMS.

[2]  Uwe H. Suhl,et al.  Supernode processing of mixed-integer models , 1994, Comput. Optim. Appl..

[3]  GAPRobert E. BixbyILOG.  MIP : THEORY AND PRACTICE { CLOSING THE , 2000 .

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

[5]  Bruce Hendrickson,et al.  Sparse Matrix Ordering Methods for Interior Point Linear Programming , 1998, INFORMS J. Comput..

[6]  A. Land,et al.  An Automatic Method for Solving Discrete Programming Problems , 1960, 50 Years of Integer Programming.

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

[8]  Egon Balas,et al.  Gomory cuts revisited , 1996, Oper. Res. Lett..

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

[10]  Martin W. P. Savelsbergh,et al.  Lifted Cover Inequalities for 0-1 Integer Programs: Complexity , 1999, INFORMS J. Comput..

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

[12]  Robert Weismantel,et al.  On the 0/1 knapsack polytope , 1997, Math. Program..

[13]  Martin W. P. Savelsbergh,et al.  Lifted flow cover inequalities for mixed 0-1 integer programs , 1999, Math. Program..

[14]  J. Gilbert,et al.  Sparse Partial Pivoting in Time Proportional to Arithmetic Operations , 1986 .

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

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

[17]  Roy E. Marsten,et al.  The Design of the XMP Linear Programming Library , 1981, TOMS.

[18]  E. Andrew Boyd,et al.  An Efficient Algorithm for Solving an Air Traffic Management Model of the National Airspace System , 1998, INFORMS J. Comput..

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

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

[21]  Jeffery L. Kennington,et al.  An Empirical Evaluation of the KORBX® Algorithms for Military Airlift Applications , 1990, Oper. Res..

[22]  Pascal Van Hentenryck.  The OPL optimization programming language , 1999 .