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We consider the problem of pattern detection in large scale
railway timetables. This problem arises in rolling stock optimization planning
in order to identify invariant sections of the timetable for
which a cyclic rotation plan is adequate.
We propose a dual reduction technique which leads to an decomposition
and enumeration method. Computational results for real
world instances demonstrate that the method is able to
produce optimal solutions as fast as standard MIP solvers.
The analysis of infeasible subproblems plays an import role in solving mixed integer programs (MIPs) and is implemented in most major MIP solvers. There are two fundamentally different concepts to generate valid global constraints from infeasible subproblems. The first is to analyze the sequence of implications obtained by domain propagation that led to infeasibility. The result of the analysis is one or more sets of contradicting variable bounds from which so-called conflict constraints can be generated. This concept has its origin in solving satisfiability problems and is similarly used in constraint programming. The second concept is to analyze infeasible linear programming (LP) relaxations. The dual LP solution provides a set of multipliers that can be used to generate a single new globally valid linear constraint. The main contribution of this short paper is an empirical evaluation of two ways to combine both approaches. Experiments are carried out on general MIP instances from standard public test sets such as Miplib2010; the presented algorithms have been implemented within the non-commercial MIP solver SCIP. Moreover, we present a pool-based approach to manage conflicts which addresses the way a MIP solver traverses the search tree better than aging strategies known from SAT solving.
The analysis of infeasible subproblems plays an import role in solving mixed integer programs (MIPs) and is implemented in most major MIP solvers. There are two fundamentally different concepts to generate valid global constraints from infeasible subproblems. The first is to analyze the sequence of implications obtained by domain propagation that led to infeasibility. The result of the analysis is one or more sets of contradicting variable bounds from which so-called conflict constraints can be generated. This concept has its origin in solving satisfiability problems and is similarly used in constraint programming. The second concept is to analyze infeasible linear programming (LP) relaxations. The dual LP solution provides a set of multipliers that can be used to generate a single new globally valid linear constraint. The main contribution of this short paper is an empirical evaluation of two ways to combine both approaches. Experiments are carried out on general MIP instances from standard public test sets such as Miplib2010; the presented algorithms have been implemented within the non-commercial MIP solver SCIP. Moreover, we present a pool-based approach to manage conflicts which addresses the way a MIP solver traverses the search tree better than aging strategies known from SAT solving.
The analysis of infeasibility plays an important role in solving satisfiability problems (SAT) and mixed integer programs (MIPs). In mixed integer programming, this procedure is called conflict analysis. So far, modern MIP solvers use conflict analysis only for propagation and improving the dual bound, i.e., fathoming nodes that cannot contain feasible solutions. In this short paper, we present a new approach which uses conflict information to improve the primal bound during a MIP solve. To derive new improving primal solutions we use a conflict driven diving heuristic called conflict diving that uses the information obtained by conflict analysis. Conflict diving pursues a twofold strategy. By using conflict information the new diving approach is guided into parts of the search space that are usually not explored by other diving heuristics. At the same time, conflict diving has a fail-fast-strategy to reduce the time spent if it cannot find a new primal solution. As a byproduct, additional valid conflict constraints can be derived, from which a MIP solver can gain benefit to improve the dual bound as well. To show the added-value of conflict diving within a MIP solver, conflict diving has been implemented within the non-commercial MIP solver SCIP. Experiments are carried out on general MIP instances from standard public test sets, like MIPLIB2010 or Cor@l.
Heutzutage ist eine Vielzahl der mehrstöckigen Gebäude mit Personenaufzugsgruppen
ausgestattet. Uns wohl bekannt sind die sogenannten konventionellen Systeme. Bei
diesen Systemen betätigt jeder ankommende Passagier eine der beiden Richtungstasten
und teilt dem dahinterstehenden Steuerungsalgorithmus seine gewünschte Startetage
und Fahrtrichtung mit. Betreten wird der zuerst auf der Startetage ankommende Aufzug
mit gleicher Fahrtrichtung und ausreichend Kapazität. Die entsprechende Zieletage
wird dem System erst nach dem Betreten der Fahrgastkabine mitgeteilt. Neben diesen
konventionellen Systemen gibt es Aufzugsgruppen mit Zielrufsteuerung. Die Besonderheit
eines zielrufgesteuerten Systems ist, dass ein ankommender Passagier bereits auf
der Startetage seine gewünschte Zieletage angibt und eine Rückmeldung vom System
erhält, welchen Aufzug er nutzen soll. Diese Zuweisung durch das System hat das Ziel,
die Warte- und Reisezeiten der Passagiere zu minimieren. Ein wesentlicher Faktor bei
der Berechnung warte- und reisezeitminimaler Fahrpläne ist das momentane Verkehrsmuster.
Eine Einteilung der Verkehrsszenarien lässt sich am besten bei Bürogebäuden
vornehmen. So ist es typisch für die Morgenstunden, dass jeder Passagier auf einer
Zugangsebene seine Fahrt beginnt und alle Passagiere die gleiche Fahrtrichtung haben.
Unter einer Zugangsebene ist z. B. der Haupteingang oder ein Parkdeck zu verstehen.
Ein weiterer wesentlicher Punkt bei Zielrufsystemen ist die Art der Zuweisung der Passagiere
durch das System. Zum einen gibt es unmittelbar zuweisende (UZ-) Systeme.
In einem UZ-System wird nach jeder Ankunft eines Passagiers eine Momentaufnahme
des momentanen Verkehrs erstellt und es findet eine Neuplanung und Zuweisung statt.
Eine solche Momentaufnahme werden wir im späteren Verkauf als Schnappschussproblem
bezeichnen. Jeder Passagier bekommt im Anschluss an die Lösung des Schnappschussproblems
eine Mitteilung vom System, z. B. über ein Display, welchen Aufzug
er benutzen soll. Zum anderen gibt es verzögert zuweisende (VZ-) Systeme. In diesen
Systemen wird die Erstellung und Lösung eines Schnappschussproblems bis kurz vor
Ankunft eines Aufzuges auf einer Etage verzögert. In einem VZ-System teilt das System
allen wartenden Passagieren die geplanten Zieletagen des ankommenden Aufzugs mit.
Jeder Passagier, der einen Ruf getätigt hat und zu einer dieser Zieletagen fahren will,
kann jetzt diesen Aufzug betreten. Durch die Verzögerung muss im Vergleich zu einem
UZ-System eine weitaus größere Menge von Passagieren zugewiesen werden. Dadurch
kann der Lösungsprozess bedeutend aufwändiger werden. Vorteil eines VZ-Systems ist
hingegen der größere Freiheitsgrad bei der Optimierung, da aufgrund der späten Zuweisung
die weitere Verkehrsentwicklung mit einbezogen werden kann.
VZ-Systeme sind aufgrund des größeren Freiheitsgrades interessant für die
Praxis ist, wir uns demzufolge in dieser Arbeit mit einer effizienteren Lösung dieser
Art von Schnappschussproblemen befassen. Es genügt dabei den Lösungsprozess eines
Schnappschussproblems zu betrachten. Das Ziel ist eine Reduzierung der benötigten
Rechenzeit. Unter Reoptimierung verstehen wir die Konstruktion
zulässiger Spalten in den jeweiligen Iterationsrunden der Spaltengenerierung
innerhalb eines Schnappschussproblems. Als eine Iterationsrunde bezeichnet wir einer
Menge zulässiger Touren mit negativen reduzierten Kosten. Eine effiziente Reoptimierung
zeichnet sich durch die Wiederverwendung und Aufbereitung von Informationen
aus vorangegangenen Iterationsrunden desselben Schnappschussproblems aus. Zu den
wichtigen Informationen gehört der konstruierte Suchbaum der vorherigen Iterationsrunde
mit seinen ausgeloteten (abgeschnittenen) Blättern sowie konstruierten Touren
bzw. Spalten, welche in der Iterationsrunde ihrer Konstruktion nicht zur Lösung des
Teilproblems der Spaltengenerierung beitrugen. Eine solche Wiederverwendung und
Aufbereitung von Informationen nennen wir Warmstart.
Recently, there have been many successful applications of optimization algorithms that solve a sequence of quite similar mixed-integer programs (MIPs) as subproblems. Traditionally, each problem in the sequence is solved from scratch. In this paper we consider reoptimization techniques that try to benefit from information obtained by solving previous problems of the sequence. We focus on the case that subsequent MIPs differ only in the objective function or that the feasible region is reduced. We propose extensions of the very complex branch-and-bound algorithms employed by general MIP solvers based on the idea to ``warmstart'' using the final search frontier of the preceding solver run. We extend the academic MIP solver SCIP by these techniques to obtain a reoptimizing branch-and-bound solver and report computational results which show the effectiveness of the approach.
State-of-the-art solvers for mixed integer programs (MIP) govern a variety of algorithmic components. Ideally, the solver adaptively learns to concentrate its computational budget on those components that perform well on a particular problem, especially if they are time consuming. We focus on three such algorithms, namely the classes of large neighborhood search and diving heuristics as well as Simplex pricing strategies. For each class we propose a selection strategy that is updated based on the observed runtime behavior, aiming to ultimately select only the best algorithms for a given instance. We review several common strategies for such a selection scenario under uncertainty, also known as Multi Armed Bandit Problem. In order to apply those bandit strategies, we carefully design reward functions to rank and compare each individual heuristic or pricing algorithm within its respective class. Finally, we discuss the computational benefits of using the proposed adaptive selection within the SCIP Optimization Suite on publicly available MIP instances.
Mixed integer nonlinear programs (MINLPs) are arguably among the hardest optimization problems, with a wide range of applications. MINLP solvers that are based on linear relaxations and spatial branching work similar as mixed integer programming (MIP) solvers in the sense that they are based on a branch-and-cut algorithm, enhanced by various heuristics, domain propagation, and presolving techniques. However, the analysis of infeasible subproblems, which is an important component of most major MIP solvers, has been hardly studied in the context of MINLPs. There are two main approaches for infeasibility analysis in MIP solvers: conflict graph analysis, which originates from artificial intelligence and constraint programming, and dual ray analysis.
The main contribution of this short paper is twofold. Firstly, we present the first computational study regarding the impact of dual ray analysis on convex and nonconvex MINLPs. In that context, we introduce a modified generation of infeasibility proofs that incorporates linearization cuts that are only locally valid. Secondly, we describe an extension of conflict analysis that works directly with the nonlinear relaxation of convex MINLPs instead of considering a linear relaxation. This is work-in-progress, and this short paper is meant to present first theoretical considerations without a computational study for that part.
Conflict learning algorithms are an important component of modern MIP and CP solvers. But strong conflict information is typically gained by depth-first search. While this is the natural mode for CP solving, it is not for MIP solving. Rapid Learning is a hybrid CP/MIP approach where CP search is applied at the root to learn information to support the remaining MIP solve. This has been demonstrated to be beneficial for binary programs. In this paper, we extend the idea of Rapid Learning to integer programs, where not all variables are restricted to the domain {0, 1}, and rather than just running a rapid CP search at the root, we will apply it repeatedly at local search nodes within the MIP search tree. To do so efficiently, we present six heuristic criteria to predict the chance for local Rapid Learning to be successful. Our computational experiments indicate that our extended Rapid Learning algorithm significantly speeds up MIP search and is particularly beneficial on highly dual degenerate problems.