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Two essential ingredients of modern mixed-integer programming (MIP) solvers are diving heuristics that simulate a partial depth-first search in a branch-and-bound search tree and conflict analysis of infeasible subproblems to learn valid constraints. So far, these techniques have mostly been studied independently: primal heuristics under the aspect of finding high-quality feasible solutions early during the solving process and conflict analysis for fathoming nodes of the search tree and improving the dual bound. Here, we combine both concepts in two different ways. First, we develop a diving heuristic that targets the generation of valid conflict constraints from the Farkas dual. We show that in the primal this is equivalent to the optimistic strategy of diving towards the best bound with respect to the objective function. Secondly, we use information derived from conflict analysis to enhance the search of a diving heuristic akin to classical coefficient diving. The computational performance of both methods is evaluated using an implementation in the source-open MIP solver SCIP. Experiments are carried out on publicly available test sets including Miplib 2010 and Cor@l.
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.
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.
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.
In this paper, we present a new, optimization-based method to exhibit cyclic behavior in non-reversible stochastic processes. While our method is general, it is strongly motivated by discrete simulations of ordinary differential equations representing non-reversible biological processes, in particular molecular simulations. Here, the discrete time steps of the simulation are often very small compared to the time scale of interest, i.e., of the whole process. In this setting, the detection of a global cyclic behavior of the process becomes difficult because transitions between individual states may appear almost reversible on the small time scale of the simulation. We address this difficulty using a mixed-integer programming model that allows us to compute a cycle of clusters with maximum net flow, i.e., large forward and small backward probability. For a synthetic genetic regulatory network consisting of a ring-oscillator with three genes, we show that this approach can detect the most productive overall cycle, outperforming classical spectral analysis methods. Our method applies to general non-equilibrium steady state systems such as catalytic reactions, for which the objective value computes the effectiveness of the catalyst.
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.
In this paper, we present a new, optimization-based method to exhibit cyclic behavior in non-reversible stochastic processes. While our method is general, it is strongly motivated by discrete simulations of ordinary differential equations representing non-reversible biological processes, in particular molecular simulations. Here, the discrete time steps of the simulation are often very small compared to the time scale of interest, i.e., of the whole process. In this setting, the detection of a global cyclic behavior of the process becomes difficult because transitions between individual states may appear almost reversible on the small time scale of the simulation. We address this difficulty using a mixed-integer programming model that allows us to compute a cycle of clusters with maximum net flow, i.e., large forward and small backward probability. For a synthetic genetic regulatory network consisting of a ring-oscillator with three genes, we show that this approach can detect the most productive overall cycle, outperforming classical spectral analysis methods. Our method applies to general non-equilibrium steady state systems such as catalytic reactions, for which the objective value computes the effectiveness of the catalyst.
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.
Many optimization problems can be modeled as Mixed Integer Programs (MIPs). In general, MIPs cannot be solved efficiently, since solving MIPs is NP-hard, see, e.g., Schrijver, 2003. Common methods for solving NP-hard problems are branch-and-bound and column generation. In the case of column generation, the original problem
becomes decomposed or re-formulated into one ore more smaller subproblems, which are easier to solve. Each of these subproblems is solved separately and recurrently, which can be interpreted as solving a sequence of optimization problems.
In this thesis, we consider a sequence of MIPs which only differ in the respective objective functions. Furthermore, we assume each of these MIPs get solved with a branch-and-bound algorithm. This thesis aims to figure out whether the solving process of a given sequence of MIPs can be accelerated by reoptimization. As reoptimization we understand starting the solving process
of a MIP of this sequence at a given frontier of a search tree corresponding to another MIP of this sequence.
At the beginning we introduce an LP-based branch-and-bound algorithm. This algorithm is inspired by the reoptimizing algorithm of Hiller, Klug, and the author of this
thesis, 2013. Since most of the state-of-the-art MIP
solvers come to decisions based on dual information, which leads to the loss of feasible solutions after changing the objective function, we present a technique to guarantee optimality despite using these information. A decision is based on a dual information if this decision is valid for at least one feasible solution, whereas a decision is based on a primal information if this decision is valid for all feasible solutions. Afterwards, we consider representing the search frontier of the tree by a set of nodes of a given size. We call this the Tree Compression Problem. Moreover, we present a criterion characterizing the similarity of two objective functions. To evaluate our approach of reoptimization we extend the well-known and well-maintained MIP solver SCIP to an LP-based branch-and-bound framework, introduce two heuristics for solving the Tree Compression Problem, and a primal heuristic which is especially fitted to column generation. Finally, we present computational experiments on several problem classes, e.g., the Vertex Coloring and k-Constrained Shortest Path. Our experiments show, that a straightforward reoptimization, i.e., without additional heuristics, provides no benefit in general. However, in combination with the techniques and methods presented in this thesis, we can accelerate the solving of a given sequence up to the factor 14. For this purpose it is essential to take the differences of the objective functions into account and to restart the reoptimization, i.e., solve the subproblem from scratch, if the objective functions are not similar enough. Finally, we discuss the possibility to parallelize the solving process of the search frontier at the beginning of each solving process.
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.
We consider reoptimization (i.e. the solution of a problem based on information available from solving a similar problem) for branch-and-bound algorithms and propose a generic framework to construct a reoptimizing branch-and-bound algorithm.
We apply this to an elevator scheduling algorithm solving similar subproblems to generate columns using branch-and-bound. Our results indicate that reoptimization techniques can substantially reduce the running times of the overall algorithm.
Conflict Analysis for MINLP
(2021)
The generalization of MIP techniques to deal with nonlinear, potentially non-convex, constraints have been a fruitful direction of research for computational MINLP in the last decade. In this paper, we follow that path in order to extend another essential subroutine of modern MIP solvers towards the case of nonlinear optimization: the analysis of infeasible subproblems for learning additional valid constraints. To this end, we derive two different strategies, geared towards two different solution approaches. These are using local dual proofs of infeasibility for LP-based branch-and-bound and the creation of nonlinear dual proofs for NLP-based branch-and-bound, respectively. We discuss implementation details of both approaches and present an extensive computational study, showing that both techniques can significantly enhance performance when solving MINLPs to global optimality.
Conflict Analysis for MINLP
(2020)
The generalization of MIP techniques to deal with nonlinear, potentially non-convex, constraints have been a fruitful direction of research for computational MINLP in the last decade. In this paper, we follow that path in order to extend another essential subroutine of modern MIP solvers towards the case of nonlinear optimization: the analysis of infeasible subproblems for learning additional valid constraints. To this end, we derive two different strategies, geared towards two different solution approaches. These are using local dual proofs of infeasibility for LP-based branch-and-bound and the creation of nonlinear dual proofs for NLP-based branch-and-bound, respectively. We discuss implementation details of both approaches and present an extensive computational study, showing that both techniques can significantly enhance performance when solving MINLPs to global optimality.
Conflict learning plays an important role in solving mixed integer programs (MIPs) and is implemented in most major MIP solvers. A major step for MIP conflict learning is to aggregate the LP relaxation of an infeasible subproblem to a single globally valid constraint, the dual proof, that proves infeasibility within the local bounds. Among others, one way of learning is to add these constraints to the problem formulation for the remainder of the search.
We suggest to not restrict this procedure to infeasible subproblems, but to also use global proof constraints from subproblems that are not (yet) infeasible, but can be expected to be pruned soon. As a special case, we also consider learning from integer feasible LP solutions. First experiments of this conflict-free learning strategy show promising results on the MIPLIB2017 benchmark set.
Two essential ingredients of modern mixed-integer programming (MIP) solvers are diving heuristics that simulate a partial depth-first search in a branch-and-bound search tree and conflict analysis of infeasible subproblems to learn valid constraints. So far, these techniques have mostly been studied independently: primal heuristics under the aspect of finding high-quality feasible solutions early during the solving process and conflict analysis for fathoming nodes of the search tree and improving the dual bound. Here, we combine both concepts in two different ways. First, we develop a diving heuristic that targets the generation of valid conflict constraints from the Farkas dual. We show that in the primal this is equivalent to the optimistic strategy of diving towards the best bound with respect to the objective function. Secondly, we use information derived from conflict analysis to enhance the search of a diving heuristic akin to classical coefficient diving. The computational performance of both methods is evaluated using an implementation in the source-open MIP solver SCIP. Experiments are carried out on publicly available test sets including Miplib 2010 and Cor@l.
Conflict learning plays an important role in solving mixed integer programs (MIPs) and is implemented in most major MIP solvers. A major step for MIP conflict learning is to aggregate the LP relaxation of an infeasible subproblem to a single globally valid constraint, the dual proof, that proves infeasibility within the local bounds. Among others, one way of learning is to add these constraints to the problem formulation for the remainder of the search.
We suggest to not restrict this procedure to infeasible subproblems, but to also use global proof constraints from subproblems that are not (yet) infeasible, but can be expected to be pruned soon. As a special case, we also consider learning from integer feasible LP solutions. First experiments of this conflict-free learning strategy show promising results on the MIPLIB2017 benchmark set.
The analysis of infeasible subproblems plays an important 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 this analysis is one or more sets of contradicting variable bounds from which so-called conflict constraints can be generated. This concept is called conflict graph analysis and 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. Every ray of the dual LP provides a set of multipliers that can be used to generate a single new globally valid linear constraint. This method is called dual proof analysis. The main contribution of this paper is twofold. Firstly, we present three enhancements of dual proof analysis: presolving via variable cancellation, strengthening by applying mixed integer rounding functions, and a filtering mechanism. Further, we provide an intense computational study evaluating the impact of every presented component regarding dual proof analysis. Secondly, this paper presents the first integrated approach to use both conflict graph and dual proof analysis simultaneously within a single MIP solution process. All experiments are carried out on general MIP instances from the standard public test set MIPLIB 2017; the presented algorithms have been implemented within the non-commercial MIP solver SCIP and the commercial MIP solver FICO Xpress.