Ted K. Ralphs

On the capacitated vehicle routing problem

Abstract. We consider the Vehicle Routing Problem, in which a fixed fleet of delivery vehicles of uniform capacity must service known customer demands for a single commodity from a common depot at minimum transit cost. This difficult combinatorial problem contains both the Bin Packing Problem and the Traveling Salesman Problem (TSP) as special cases and conceptually lies at the intersection of these two well-studied problems. The capacity constraints of the integer programming formulation of this routing model provide the link between the underlying routing and packing structures. We describe a decomposition-based separation methodology for the capacity constraints that takes advantage of our ability to solve small instances of the TSP efficiently. Specifically, when standard procedures fail to separate a candidate point, we attempt to decompose it into a convex combination of TSP tours; if successful, the tours present in this decomposition are examined for violated capacity constraints; if not, the Farkas Theorem provides a hyperplane separating the point from the TSP polytope. We present some extensions of this basic concept and a general framework within which it can be applied to other combinatorial models. Computational results are given for an implementation within the parallel branch, cut, and price framework SYMPHONY.

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.

Parallel branch and cut for capacitated vehicle routing

Combinatorial optimization problems arise commonly in logistics applications. The most successful approaches to date for solving such problems involve modeling them as integer programs and then applying some variant of the branch and bound algorithm. Although branch and bound is conceptually easy to parallelize, achieving scalability can be a challenge. In more sophisticated variants, such as branch and cut, large amounts of data must be shared among the processors, resulting in increased parallel overhead. In this paper, we review the branch and cut algorithm for solving combinatorial optimization problems and describe the implementation of SYMPHONY, a library for implementing these algorithms in parallel. We then describe a solver for the vehicle routing problem that was implemented using SYMPHONY and analyze its parallel performance on a Beowulf cluster.

Parallel branch, cut, and price for large-scale discrete optimization

Abstract.In discrete optimization, most exact solution approaches are based on branch and bound, which is conceptually easy to parallelize in its simplest forms. More sophisticated variants, such as the so-called branch, cut, and price algorithms, are more difficult to parallelize because of the need to share large amounts of knowledge discovered during the search process. In the first part of the paper, we survey the issues involved in parallelizing such algorithms. We then review the implementation of SYMPHONY and COIN/BCP, two existing frameworks for implementing parallel branch, cut, and price. These frameworks have limited scalability, but are effective on small numbers of processors. Finally, we briefly describe our next-generation framework, which improves scalability and further abstracts many of the notions inherent in parallel BCP, making it possible to implement and parallelize more general classes of algorithms.

A Branch-and-cut Algorithm for Integer Bilevel Linear Programs

We describe a rudimentary branch-and-cut algorithm for solving integer bilevel linear programs that extends existing techniques for standard integer linear programs to this very challenging computational setting. The algorithm improves on the branch-and-bound algorithm of Moore and Bard in that it uses cutting plane techniques to produce improved bounds, does not require specialized branching strategies, and can be implemented in a straightforward way using only linear solvers. An implementation built using software components available in the COIN-OR software repository is described and preliminary computational results presented.

An improved algorithm for solving biobjective integer programs

A parametric algorithm for identifying the Pareto set of a biobjective integer program is proposed. The algorithm is based on the weighted Chebyshev (Tchebycheff) scalarization, and its running time is asymptotically optimal. A number of extensions are described, including: a technique for handling weakly dominated outcomes, a Pareto set approximation scheme, and an interactive version that provides access to all Pareto outcomes. Extensive computational tests on instances of the biobjective knapsack problem and a capacitated network routing problem are presented.

The Symphony Callable Library for Mixed Integer Programming

SYMPHONY is a customizable, open-source library for solving mixed-integer linear programs (MILP) by branch, cut, and price. With its large assortment of parameter settings, user callback functions, and compile-time options, SYMPHONY can be configured as a generic MILP solver or an engine for solving difficult MILPs by means of a fully customized algorithm. SYMPHONY can run on a variety of architectures, including single-processor, distributed-memory parallel, and shared-memory parallel architectures under MS Windows, Linux, and other Unix operating systems. The latest version is implemented as a callable library that can be accessed either through calls to the native C application program interface, or through a C++ interface class derived from the COIN-OR Open Solver Interface. Among its new features are the ability to solve bicriteria MILPs, the ability to stop and warm start MILP computations after modifying parameters or problem data, the ability to create persistent cut pools, and the ability to perform rudimentary sensitivity analysis on MILPs.

A Branch-and-Price Algorithm for Combined Location and Routing Problems Under Capacity Restrictions

We investigate the problem of simultaneously determining the location of facilities and the design of vehicle routes to serve customer demands under vehicle and facility capacity restrictions. We present a set-partitioning-based formulation of the problem and study the relationship between this formulation and the graph-based formulations that have been used in previous studies of this problem. We describe a branch-and-price algorithm based on the set-partitioning formulation and discuss computational experience with both exact and heuristic variants of this algorithm.

A Conic Representation of the Convex Hull of Disjunctive Sets and Conic Cuts for Integer Second Order Cone Optimization

We study the convex hull of the intersection of a convex set E and a disjunctive set. This intersection is at the core of solution techniques for Mixed Integer Convex Optimization. We prove that if there exists a cone K (resp., a cylinder C) that has the same intersection with the boundary of the disjunction as E, then the convex hull is the intersection of E with K (resp., C).The existence of such a cone (resp., a cylinder) is difficult to prove for general conic optimization. We prove existence and unicity of a second order cone (resp., a cylinder), when E is the intersection of an affine space and a second order cone (resp., a cylinder). We also provide a method for finding that cone, and hence the convex hull, for the continuous relaxation of the feasible set of a Mixed Integer Second Order Cone Optimization (MISOCO) problem, assumed to be the intersection of an ellipsoid with a general linear disjunction. This cone provides a new conic cut for MISOCO that can be used in branch-and-cut algorithms for MISOCO problems.

Computational Experience with a Software Framework for Parallel Integer Programming

In this paper, we discuss the challenges that arise in parallelizing algorithms for solving generic mixed integer linear programs and introduce a software framework that aims to address these challenges. Although the framework makes few algorithmic assumptions, it was designed specifically with support for implementation of relaxation-based branch-and-bound algorithms in mind. Achieving efficiency for such algorithms is particularly challenging and involves a careful analysis of the trade-offs inherent in the mechanisms for sharing the large amounts of information that can be generated. We present computational results that illustrate the degree to which various sources of parallel overhead affect scalability and discuss why properties of the problem class itself can have a substantial effect on the efficiency of a particular methodology.