Robert E. Bixby

Solving Real-World Linear Programs: A Decade and More of Progress

This paper is an invited contribution to the 50th anniversary issue of the journalOperations Research, published by the Institute of Operations Research and Management Science (INFORMS). It describes one persons perspective on the development of computational tools for linear programming. The paper begins with a short personal history, followed by historical remarks covering the some 40 years of linear-programming developments that predate my own involvement in this subject. It concludes with a more detailed look at the evolution of computational linear programming since 1987.

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.

MIPLIB 2010 - Mixed Integer Programming Library version 5

This paper reports on the fifth version of the Mixed Integer Programming Library. The miplib 2010 is the first miplib release that has been assembled by a large group from academia and from industry, all of whom work in integer programming. There was mutual consent that the concept of the library had to be expanded in order to fulfill the needs of the community. The new version comprises 361 instances sorted into several groups. This includes the main benchmark test set of 87 instances, which are all solvable by today’s codes, and also the challenge test set with 164 instances, many of which are currently unsolved. For the first time, we include scripts to run automated tests in a predefined way. Further, there is a solution checker to test the accuracy of provided solutions using exact arithmetic.

Very large-scale linear programming: a case study in combining interior point and simplex methods

Experience with solving a 12.753.313 variable linear program is described. This problem is the linear programming relaxation of a set partitioning problem arising from an airline crew scheduling application. A scheme is described that requires successive solutions of small subproblems, yielding a procedure that has little growth in solution time in terms of the number of variables. Experience using the simplex method as implemented in CPLEX, an interior point method as implemented in OBI, and a hybrid interior point/simplex approach is reported. The resulting procedure illustrates the power of an interior point/simplex combination for solving very large-scale linear programs.

Implementing the Dantzig-Fulkerson-Johnson algorithm for large traveling salesman problems

Abstract. Dantzig, Fulkerson, and Johnson (1954) introduced the cutting-plane method as a means of attacking the traveling salesman problem; this method has been applied to broad classes of problems in combinatorial optimization and integer programming. In this paper we discuss an implementation of Dantzig et al.s method that is suitable for TSP instances having 1,000,000 or more cities. Our aim is to use the study of the TSP as a step towards understanding the applicability and limits of the general cutting-plane method in large-scale applications.

TSP Cuts Which Do Not Conform to the Template Paradigm

The first computer implementation of the Dantzig-Fulkerson-Johnson cutting-plane method for solving the traveling salesman problem, written by Martin, used subtour inequalities as well as cutting planes of Gomorys type. The practice of looking for and using cuts that match prescribed templates in conjunction with Gomory cuts was continued in computer codes of Miliotis, Land, and Fleischmann. Grotschel, Padberg, and Hong advocated a different policy, where the template paradigm is the only source of cuts; furthermore, they argued for drawing the templates exclusively from the set of linear inequalities that induce facets of the TSP polytope. These policies were adopted in the work of Crowder and Padberg, in the work of Grotschel and Holland, and in the work of Padberg and Rinaldi; their computer codes produced the most impressive computational TSP successes of the nineteen eighties. Eventually, the template paradigm became the standard frame of reference for cutting planes in the TSP. The purpose of this paper is to describe a technique for finding cuts that disdains all understanding of the TSP polytope and bashes on regardless of all prescribed templates. Combining this technique with the traditional template approach was a crucial step in our solutions of a 13,509-city TSP instance and a 15,112-city TSP instance.

An almost linear-time algorithm for graph realization

Given a {0, 1}-matrix M, the graph-realization problem for M is to find a tree T such that the columns of M are incidence vectors of paths in T, or to show that no such T exists. An algorithm is presented for this problem the time complexity of which is very nearly linear in the number of ones in M.

The Partial Order of a Polymatroid Extreme Point

Given a (polymatroid) rank function f and its corresponding polymatroid P(f), we associate with each extreme point of P(f) a certain partial order. We show that this partial order is efficiently constructible, and that it characterizes all the orderings with which the greedy algorithm can be used to generate the given extreme point. We give several applications, including one to the still open problem of finding an efficient combinatorial procedure for testing membership in polymatroids. Our results can also be applied to convex games.

A note on detecting simple redundancies in linear systems

Two efficient algorithms are presented that, for a given linear system Ax = b, eliminate equations that are non-zero multiples of other equations. The second algorithm runs in linear time when the entries of A are +1, -1 or 0.

Recovering an optimal LP basis from an interior point solution

An important issue in the implementation of interior point algorithms for linear programming is the recovery of an optimal basic solution from an optimal interior point solution. In this paper we describe a method for recovering such a solution. Our implementation links a high-performance interior point code (OB1) with a high-performance simplex code (CPLEX). Results of our computational tests indicate that basis recovery can be done quickly and efficiently.