Solving the Train Dispatching Problem in Large Networks by Column Generation

Disruptions in the operational flow of rail traffic can lead to conflicts between train movements, such that a scheduled timetable can no longer be realised. This is where dispatching is applied, existing conflicts are resolved and a dispatching timetable is provided. In the process, train paths are varied in their spatio-temporal course. This is called the train dispatching problem (TDP), which consists of selecting conflict-free train paths with minimum delay. Starting from a path-oriented formulation of the TDP, a binary linear decision model is introduced. For each possible train path, a binary decision variable indicates whether the train path is used by the request or not. Such a train path is constructed from a set of predefined path parts (\profiles{}) within a time-space network. Instead of modelling pairwise conflicts, stronger MIP formulation are achieved by a clique formulation. The combinatorics of speed profiles and different departure times results in a large number of possible train paths, so that the column generation method is used here. New train paths within the pricing-problem can be calculated using shortest path techniques. Here, the shadow prices of conflict cliques must be taken into account. When constructing a new train path, it must be determined whether this train path belongs to a clique or not. This problem is tackled by a MIP. The methodology is tested on practical size instances from a dispatching area in Germany. Numerical results show that the presented method achieves acceptable computation times with good solution quality while meeting the requirements for real-time dispatching.Comment: 12 pages, 4 figures, 2 table

Small and strong formulations for unions of convex sets from the Cayley embedding

There is often a significant trade-off between formulation strength and size in mixed integer programming (MIP). When modelling convex disjunctive constraints (e.g. unions of convex sets) this trade-off can be resolved by adding auxiliary continuous variables. However, adding these variables can result in a deterioration of the computational effectiveness of the formulation. For this reason, there has been considerable interest in constructing strong formulations that do not use continuous auxiliary variables. We introduce a technique to construct formulations without these detrimental continuous auxiliary variables. To develop this technique we introduce a natural nonpolyhedral generalization of the Cayley embedding of a family of polytopes and show it inherits many geometric properties of the original embedding. We then show how the associated formulation technique can be used to construct small and strong formulation for a wide range of disjunctive constraints. In particular, we show it can recover and generalize all known strong formulations without continuous auxiliary variables.National Science Foundation (U.S.) (grant CMMI-1351619

Almost Symmetries and the Unit Commitment Problem

This thesis explores two main topics. The first is almost symmetry detection on graphs. The presence of symmetry in combinatorial optimization problems has long been considered an anathema, but in the past decade considerable progress has been made. Modern integer and constraint programming solvers have automatic symmetry detection built-in to either exploit or avoid symmetric regions of the search space. Automatic symmetry detection generally works by converting the input problem to a graph which is in exact correspondence with the problem formulation. Symmetry can then be detected on this graph using one of the excellent existing algorithms; these are also the symmetries of the problem formulation.The motivation for detecting almost symmetries on graphs is that almost symmetries in an integer program can force the solver to explore nearly symmetric regions of the search space. Because of the known correspondence between integer programming formulations and graphs, this is a first step toward detecting almost symmetries in integer programming formulations. Though we are only able to compute almost symmetries for graphs of modest size, the results indicate that almost symmetry is definitely present in some real-world combinatorial structures, and likely warrants further investigation.The second topic explored in this thesis is integer programming formulations for the unit commitment problem. The unit commitment problem involves scheduling power generators to meet anticipated energy demand while minimizing total system operation cost. Today, practitioners usually formulate and solve unit commitment as a large-scale mixed integer linear program.The original intent of this project was to bring the analysis of almost symmetries to the unit commitment problem. Two power generators are almost symmetric in the unit commitment problem if they have almost identical parameters. Along the way, however, new formulations for power generators were discovered that warranted a thorough investigation of their own. Chapters 4 and 5 are a result of this research.Thus this work makes three contributions to the unit commitment problem: a convex hull description for a power generator accommodating many types of constraints, an improved formulation for time-dependent start-up costs, and an exact symmetry reduction technique via reformulation

On the interplay of Mixed Integer Linear, Mixed Integer Nonlinear and Constraint Programming

In this thesis we study selected topics in the field of Mixed Integer Programming (MIP), in particular Mixed Integer Linear and Nonlinear Programming (MI(N)LP). We set a focus on the influences of Constraint Programming (CP). First, we analyze Mathematical Programming approaches to water network optimization, a set of challenging optimization problems frequently modeled as non-convex MINLPs. We give detailed descriptions of many variants and survey solution approaches from the literature. We are particularly interested in MILP approximations and present a respective computational study for water network design problems. We analyze this approach by algorithmic considerations and highlight the importance of certain convex substructures in these non-convex MINLPs. We further derive valid inequalities for water network design problems exploiting these substructures. Then, we treat Mathematical Programming problems with indicator constraints, recalling their most popular reformulation techniques in MIP, leading to either big-M constraints or disjunctive programming techniques. The latter give rise to reformulations in higher-dimensional spaces, and we review special cases from the literature that allow to describe the projection on the original space of variables explicitly. We theoretically extend the respective results in two directions and conduct computational experiments. We then present an algorithm for MILPs with indicator constraints that incorporates elements of CP into MIP techniques, including computational results for the JobShopScheduling problem. Finally, we introduce an extension of the class of MILPs so that linear expressions are allowed to have non-contiguous domains. Inspired by CP, this permits to model holes in the domains of variables as a special case. For such problems, we extend the theory of split cuts and show two ways of separating them, namely as intersection and lift-and-project cuts, and present computational results. We further experiment with an exact algorithm for such problems, applied to the Traveling Salesman Problem with multiple time windows

On mathematical programming with indicator constraints

In this paper we review the relevant literature on mathematical optimiza- tion with logical implications, i.e., where constraints can be either active or disabled depending on logical conditions to hold. In the case of convex functions, the theory of disjunctive programming allows one to formulate these logical implications as convex nonlinear programming problems in a space of variables lifted with respect to its origi- nal dimension. We concentrate on the attempt of avoiding the issue of dealing with large NLPs. In particular, we review some existing results that allow to work in the original space of variables for two relevant special cases where the disjunctions corresponding to the logical implications have two terms. Then, we significantly extend these spe- cial cases in two different directions, one involving more general convex sets and the other with disjunctions involving three terms. Computational experiments comparing disjunctive programming formulations in the original space of variables with straight- forward bigM ones show that the former are computationally viable and promising