Aspects of Duality in Integer Programming

This thesis deals with various problems arising when dualizing integer programs and combinatorial optimization problems. On the one hand, the corresponding dual functions are piecewise linear, calling for approaches from nondifferentiable optimization; on the other hand, the problems have special structures which can be exploited. Line searches along the dual function of integer programs are an example of when the latter is the case. We introduce efficient specialized line search procedures for the dual functions of binary programs, the Lagrangian-surrogate dual of the TSP and the p-median problem, and for the dual function of the shortest path problem with a knapsack constraint. A specialized line search is also used for a conditional epsilon-ascent bundle method for the dual function of binary programs. There the line search has a different role than before: instead of finding the maximum of a one-dimensional function, the line search must deliver a conditional epsilon-subgradient. We show how this can be done efficiently and exactly. We employ duality theory to the satisfiability problem (SAT) in order to prove unsatisfiability. In order to do this, the SAT problem is converted into an equivalent binary program modeling the "minimum unsatisfiable clauses" problem. We approach this problem by dualizing the problem, yielding a lower bound on the number of unsatisfiable clauses. If this value is strictly positive, then the instance is nonsatisfiable. Two dualizations are tested, namely the Lagrangian dual (with very moderate success) and the split dual from Lagrangian decomposition (variable splitting), with encouraging results. In a final note we discuss possibilities to extend the usual notion of piecewise linear in order to capture functions which "look" piecewise linear, but wouldn\u27t fit the usual definition. We comment on certain difficulties one encounters when doing this

Overview Over Optimization Models in Transportation

this paper with options improving the ability of the presented models to scale in terms of problem size and in terms of better handling of complicated resource constraints

High Performance Integer Optimization for Crew Scheduling

Performance aspects of a Lagrangian relaxation based heuristic for solving large 0-1 integer linear programs are discussed. In particular, we look at its application to airline and railway crew scheduling problems. We present a scalable parallelization of the original algorithm used in production at Carmen Systems AB, Goteborg, Sweden, based on distributing the variables and a new sequential active set strategy which requires less work and is better adapted to the memory hierachy properties of modern RISC processors. The active set strategy can even be parallelized on networks of workstations

Parallel crew scheduling in PAROS

. We give an overview of the parallelization work done in PAROS. The specific parallelization objective has been to improve the speed of airline crew scheduling, on a network of workstations. The work is based on the Carmen System, which is used by most European airlines for this task. We give a brief background to the problem. The two most time critical parts of this system are the pairing generator and the optimizer. We present a pairing generator which distributes the enumeration of pairings over the processors. This works efficiently on a large number of loosely coupled workstations. The optimizer, which is based on an iterative Lagrangian heuristic, allows only for rather fine-grained parallelization. On low-latency machines, parallelizing the two innermost loops at once works well. A new "active-set" strategy makes more coarse-grained communication possible and even improves the sequential algorithm. 1 The PAROS project The PAROS (Parallel large scale automatic scheduling) proje..

A Fair Treatment For All Growers At The Auction

This paper deals with the question, whether the current procedure flowers are auctioned at a major dutch auctioning centre can be considered as fair or not from the growers view. Some growers have stated, that this is not the case. We first introduce a notion of "fairness", which is meant to approximate the real feeling of a fair or unfair treatment. Then we compare the current procedure used at the auctioning centre with other procedures which we developed and present their effect on our approximation of fairness in a simulation environment. We conclude that the auctioning centre should revise their auctioning system, since it shows a significant bias in the growers' chances to sell their products. We show how this drawback can be overcome with. 1 Introduction Flowers Auction is an international trading center with auctioning centers in Naaldwijk and Bleiswijk, the Netherlands. In these auctioning centers a wide range of high quality flowers and plants are traded. The hear..