... In this paper, we survey the various theoretical and practical issues related to degeneracy in IPM's for linear programming. We survey results which for the most part already appeared in the literature. Roughly speaking, we shall deal with four topics: the effect of degeneracy on the convergence of IPM's, on the trajectories followed by the algorithms, the effect of degeneracy in numerical performance, and on finding basic solutions.

We describe a new exact-arithmetic approach to linear programming when the number of variables n is much larger than the number of constraints m (or vice versa). The algorithm is an implementation of the simplex method which combines exact (multiple precision) arithmetic with inexact (floating point) arithmetic, where the number of exact arithmetic operations is small and usually bounded by a function of min(n; m). Combining this with a "partial pricing" scheme (based on a result by Clarkson [8]) which is particularly tuned for the problems under consideration, we obtain a correct and practically efficient algorithm that even competes with the inexact state-of-the-art solver CPLEX 1 for small values of min(n; m) and and is far superior to methods that use exact arithmetic in any operation. 1 Introduction Linear Programming (LP) -- the problem of maximizing a linear objective function in n variables subject to m linear (in)equality constraints -- is the most prominent optimization ...

This paper outlines an alternative solution method which has been incorporated into a system which originated from IMPACS. Improved results on a selection of real bus driver problems are presented. THE DRIVER SCHEDULING PROBLEM

The purpose of this paper is to survey the various pivot rules of the simplex method or its variants that have been developed in the last two decades, starting from the appearance of the minimal index rule of Bland. We are mainly concerned with the finiteness property of simplex type pivot rules. There are some other important topics in linear programming, e.g. complexity theory or implementations, that are not included in the scope of this paper. We do not discuss ellipsoid methods nor interior point methods. Well known classical results concerning the simplex method are also not particularly discussed in this survey, but the connection between the new methods and the classical ones are discussed if there is any. In this paper we discuss three classes of recently developed pivot rules for linear programming. The first class (the largest one) of the pivot rules we discuss is the class of essentially combinatorial pivot rules. Namely these rules only use labeling and signs of the variab...

We study the parallelization of the steepest-edge version of the dual simplex

In this paper we provide a compact, formal description for an important class of problems involving the deployment of diverse equipment (e.g., planes, ships, trucks) to transport cargo (e.g., personnel and material) throughout large geographical areas for private, government, and military purposes. The problem is particularly interesting for its combination of static and dynamic sources of complexity, and the fact that it allows and in some cases strongly encourages distributed solutions. The merit of this formalization is shown by its application to both understanding the complexities in existing restricted transportation problems and creating new natural restrictions on the general problem. Starting from the general formulation we are able to isolate the static and dynamic sources of complexity in transportation problems and relate them to existing research in computer science and operations research. In addition, we describe techniques for constructing algorithms for the transportat...

This paper discusses a decision support system for airline and railway crew planning. The system is a state-of-the-art branch-and-price solver that is used for crew scheduling and crew rostering. We briefly discuss the mathematical background of the solver, of which most part is covered in the Operations Research literature. Crew scheduling is crew planning for one or a few days that results in crew duties or pairings, and crew rostering is crew planning for at least one week for individual crew members. Technical issues about the system and its implementation are covered in more detail, as well as several applications. In particular, we focus on a specific aircrew rostering application. The computational results contain an interesting comparison of results obtained with, on one hand, the approach in which crew scheduling is carried out before crew rostering, and, on the other hand, an approach in which these two planning problems are solved in an integrated manner.

Random sampling is an efficient method to deal with constrained optimization problems in computational geometry. In a first step, one finds the optimal solution subject to a random subset of the constraints; in many cases, the expected number of constraints still violated by that solution is then significantly smaller than the overall number of constraints that remain. This phenomenon can be exploited in several ways, and typically results in simple and asymptotically fast algorithms. Very often the analysis of random sampling in this context boils down to a simple identity (the sampling lemma) which holds in an amazingly general framework, yet has not explicitly been stated in the literature. In the more restricted but still general setting of LP-type problems, we prove tail estimates for the sampling lemma, giving Chernoff-type bounds for the number of constraints violated by the solution of a random subset. As an application, we provide the first theoretical analysis of multiple p...

: We present a cutting plane algorithm for the feasibility problem that uses a homogenized self-dual approach to regain an approximate center when adding a cut. The algorithm requires a fully polynomial number of Newton steps. One novelty in the analysis of the algorithm is the use of a powerful proximity measure which is widely used in interior point methods but not previously used in the analysis of cutting plane methods. Moreover, a practical implementation of a variant of the homogenized cutting plane for solution of LPs is presented. Computational results with this implementation show that it is possible to solve a problem having several thousand constraints and about one million variables on a standard PC in a moderate amount of time. Key words: Interior-point, Homogeneous, cutting plane, set-partitioning. Permanent address: Math Sciences, RPI, Troy NY 12180, USA. Supported in part by fellowships from NWO and TU Delft 1 2 1 INTRODUCTION We are trying to nd a point y in a...

Thanks to my family; my father Robert and my mother Johanna, and my sisters Maryke and Annaleen, for their encouragement, correspondence and visits, and for having me in their thoughts so often. It is their love and support through the years that made this achievement possible. ii