90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING
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We provide an overview of new theoretical results that we obtained while further investigating multiband robust optimization, a new model for robust optimization that we recently proposed to tackle uncertainty in mixed-integer linear programming. This new model extends and refines the classical Gamma-robustness model of Bertsimas and Sim and is particularly useful in the common case of arbitrary asymmetric distributions of the uncertainty. Here, we focus on uncertain 0-1 programs and we analyze their robust counterparts when the uncertainty is represented through a multiband set. Our investigations were inspired by the needs of our industrial partners in the research project ROBUKOM.
We show that a class of semidefinite programs (SDP) admits a solution that is a positive semidefinite
matrix of rank at most $r$, where $r$ is the rank of the matrix involved in the objective function of the SDP.
The optimization problems of this class are semidefinite packing problems,
which are the SDP analogs to vector packing problems.
Of particular interest is the case in which our result guarantees the existence of a solution
of rank one: we show that the computation of this solution actually reduces to a
Second Order Cone Program (SOCP).
We point out an application in statistics, in the optimal design of experiments.
We propose a new coarse-to-fine approach to solve certain linear programs by column generation. The problems that we address contain layers corresponding to different levels of detail, i.e., coarse layers as well as fine layers. These layers are utilized to design
efficient pricing rules. In a nutshell, the method shifts the pricing of a fine linear program to a coarse counterpart. In this way, major decisions are taken in the coarse layer, while minor
details are tackled within the fine layer. We elucidate our methodology by an application to a complex railway rolling stock rotation problem. We provide comprehensive computational results that demonstrate the benefit of this new technique for the solution of large scale problems.
The last milestone achievement for the roundoff-error-free solution of general mixed integer programs over the rational numbers was a hybrid-precision branch-and-bound algorithm published by Cook, Koch, Steffy, and Wolter in 2013. We describe a substantial revision and extension of this framework that integrates symbolic presolving, features an exact repair step for solutions from primal heuristics, employs a faster rational LP solver based on LP iterative refinement, and is able to produce independently verifiable certificates of optimality. We study the significantly improved performance and give insights into the computational behavior of the new algorithmic components. On the MIPLIB 2017 benchmark set, we observe an average speedup of 6.6x over the original framework and 2.8 times as many instances solved within a time limit of two hours.
We introduce a concurrent solver for the periodic event scheduling problem (PESP). It combines mixed integer programming techniques, the modulo network simplex method, satisfiability approaches, and a new heuristic based on maximum cuts. Running these components in parallel speeds up the overall solution process. This enables us to significantly improve the current upper and lower bounds for all benchmark instances of the library PESPlib.
The current cut selection algorithm used in mixed-integer programming solvers has remained largely unchanged since its creation. In this paper, we propose a set of new cut scoring measures, cut filtering techniques, and stopping criteria, extending the current state-of-the-art algorithm and obtaining a 5\% performance improvement for SCIP over the MIPLIB 2017 benchmark set.
Scheduling ist ein wichtiger Forschungsgegenstand im Bereich der diskreten Optimierung. Es geht darum, einen Schedule, d.h. einen Ablaufplan, für gegebene Ereignisse zu finden. Dieser soll optimal hinsichtlich einer Zielfunktion wie zum Beispiel minimaler Dauer oder Kosten sein. Dabei gibt es in der Regel Nebenbedingungen wie Vorrangbeziehungen zwischen den Ereignissen oder zeitliche Einschränkungen, die zu erfüllen sind. Falls die Ereignisse periodisch wiederkehren, spricht man von periodischem Scheduling. Beispiele sind das Erstellen von Zugfahrplänen, die Schaltungvon Ampelsignalen oder die Planung von Produktionsabläufen.
Mathematisch können diese Probleme mit dem Periodic Event Scheduling Problem (PESP) modelliert werden, das als gemischt-ganzzahliges Programm formuliert werden kann. In dieser Bachelorarbeit wird ein Ansatz zur Lösung des PESP mittels Zerlegung und Dualisierung entwickelt. In den Kapiteln 2 und 3 werden zunächst die notwendigen graphentheoretischen Grundlagen und das PESP eingeführt. In Kapitel 4 wird das PESP durch Fixierung der ganzzahligen Variablen in lineare Programme zerlegt. Dieses Unterproblem wird dualisiert und wieder in das PESP eingesetzt. Dafür ist eine weitere Nebenbedingung nötig.
Im fünften Kapitel behandeln wir die Lösung des teildualisierten PESP. Eine Möglichkeit ist es, sich auf eine Teilmenge der Nebenbedingungen zu beschränken. Eine weitere Möglichkeit ist ein Algorithmus, derähnlich wie BendersZerlegung die Nebenbedingungen dynamisch erzeugt. Dieser Algorithmus wird in Kapitel 6 implementiert und an vier Beispielen getestet.
Two fundamental mathematical formulations for railway timetabling are compared on a common set of sample problems, representing both multiple track high density services in Europe and single track bidirectional operations in North America. One formulation, ACP, enforces against conflicts by constraining time intervals between trains, while the other formulation, HGF, monitors physical occupation of controlled track segments. The results demonstrate that both ACP and HGF return comparable solutions in the aggregate, with some significant differences in select instances, and a pattern of significant differences in performance and constraint enforcement overall.
We consider a system dynamics model that describes the effect of human activity on natural resources. The central stocks are the accumulated profit, the industry structures, and the water resources. The model can be controlled through two time-dependent parameters. The goal in this paper is to find a parameter setting that leads to a maximization of a performance index, which reflects both environmental and economic aspects. Thus, the goal is to identify the most sustainable stock of industry structures within the model's constraints and assumptions. In order to find a proven global optimal parameter set, we formulate the System Dynamics Optimization model as a mixed-integer nonlinear problem that is accessible for numerical solvers. Due to the dynamic structure of the model, certain steps of the solution process must be handled with greater care, compared to standard non-dynamic problems. We describe our approach of solving the industry structure model and present computational results. In addition, we discuss the limitations of the approach and next steps.
The System Dynamics (SD) methodology is a framework for modeling and simulating the dynamic behavior of socioeconomic systems. Characteristic for the description of such systems is the occurrence of feedback loops together with stocks and flows. The mathematical equations that describe the system are usually ordinary differential equations and nonlinear algebraic constraints. Therefore seemingly simple systems can show a nonintuitive, unpredictable behavior over time. Controlling a dynamical system means to specify potential interventions from outside that should keep the system on the desired track, and to define an evaluation schema to compare different controls among each other, so that a "best" control can be defined in a meaningful way. The central question is how to compute such globally optimal control for a given SD model, that allows the transition of the system into a desired state with minimum effort. We propose a mixed-integer nonlinear programming (MINLP) reformulation of the System Dynamics Optimization (SDO) problem. MINLP problems can be solved by linear programming based branch-and-bound approach. We demonstrate that standard MINLP solvers are not able to solve SDO problem. To overcome this obstacle, we introduce a special-tailored bound propagation method. We apply our new method to a predator-prey model with additional hunting activity as control, and to a mini-world model with the consumption level as control. Numerical results for these test cases are presented.