An Algorithm for Biobjective Mixed Integer Quadratic Programs

Multiobjective quadratic programs (MOQPs) are appealing since convex quadratic programs have elegant mathematical properties and model important applications. Adding mixed-integer variables extends their applicability while the resulting programs become global optimization problems. Thus, in this work, we develop a branch and bound (BB) algorithm for solving biobjective mixed-integer quadratic programs (BOMIQPs). An algorithm of this type does not exist in the literature. The algorithm relies on five fundamental components of the BB scheme: calculating an initial set of efficient solutions with associated Pareto points, solving node problems, fathoming, branching, and set dominance. Considering the properties of the Pareto set of BOMIQPs, two new fathoming rules are proposed. An extended branching module is suggested to cooperate with the node problem solver. A procedure to make the dominance decision between two Pareto sets with limited information is proposed. This set dominance procedure can eliminate the dominated points and eventually produce the Pareto set of the BOMIQP. Numerical examples are provided. Solving multiobjective quadratic programs (MOQPs) is fundamental to our research. Therefore, we examine the algorithms for this class of problems with different perspectives. The scalarization techniques for (strictly) convex MOPs are reviewed and the available algorithms for computing efficient solutions for MOQPs are discussed. These algorithms are compared with respect to four properties of MOQPs. In addition, methods for solving parametric multiobjective quadratic programs are studied. Computational studies are provided with synthetic instances, and examples in statistics and portfolio optimization. The real-life context reveals the interplay between the scalarizations and provides an additional insight into the obtained parametric solution sets

Solution Techniques for Classes of Biobjective and Parametric Programs

Mathematical optimization, or mathematical programming, has been studied for several decades. Researchers are constantly searching for optimization techniques which allow one to de-termine the ideal course of action in extremely complex situations. This line of scientific inquiry motivates the primary focus of this dissertation — nontraditional optimization problems having either multiple objective functions or parametric input. Utilizing multiple objective functions al-lows one to account for the fact that the decision process in many real-life problems in engineering, business, and management is often driven by several conflicting criteria such as cost, performance, reliability, safety, and productivity. Additionally, incorporating parametric input allows one to ac-count for uncertainty in models’ data, which can arise for a number of reasons, including a changing availability of resources, estimation or measurement errors, or implementation errors caused by stor-ing data in a fixed precision format. However, when a decision problem has either parametric input or multiple objectives, one cannot hope to find a single, satisfactory solution. Thus, in this work we develop techniques which can be used to determine sets of desirable solutions. The two main problems we consider in this work are the biobjective mixed integer linear program (BOMILP) and the multiparametric linear complementarity problem (mpLCP). BOMILPs are optimization problems in which two linear objectives are optimized over a polyhedron while restricting some of the decision variables to be integer. We present a new data structure in the form of a modified binary tree that can be used to store the solution set of BOMILP. Empirical evidence is provided showing that this structure is able to store these solution sets more efficiently than other data structures that are typically used for this purpose. We also develop a branch-and-bound (BB) procedure that can be used to compute the solution set of BOMILP. Computational experiments are conducted in order to compare the performance of the new BB procedure with current state-of-the-art methods for determining the solution set of BOMILP. The results provide strong evidence of the utility of the proposed BB method. We also present new procedures for solving two variants of the mpLCP. Each of these proce-dures consists of two phases. In the first phase an initial feasible solution to mpLCP which satisfies certain criteria is determined. This contribution alone is significant because the question of how such an initial solution could be generated was previously unanswered. In the second phase the set of fea-sible parameters is partitioned into regions such that the solution of the mpLCP, as a function of the parameters, is invariant over each region. For the first variant of mpLCP, the worst-case complex-ity of the presented procedure matches that of current state-of-the-art methods for nondegenerate problems and is lower than that of current state-of-the-art methods for degenerate problems. Addi-tionally, computational results show that the proposed procedure significantly outperforms current state-of-the-art methods in practice. The second variant of mpLCP we consider was previously un-solved. In order to develop a solution strategy, we first study the structure of the problem in detail. This study relies on the integration of several key concepts from algebraic geometry and topology into the field of operations research. Using these tools we build the theoretical foundation necessary to solve the mpLCP and propose a strategy for doing so. Experimental results indicate that the presented solution method also performs well in practice

New Multiobjective Shortest Path Algorithms

The main problem studied in this thesis is the Multiobjective Shortest Path (MOSP) problem. We focus on the problem variant with three or more objectives. It is a generalization of the classical Shortest Path problem in which arcs in the input graph are weighted with vectors instead of scalars. The main contribution is the Multiobjective Dijkstra Algorithm (MDA), a label- setting algorithm that achieves state of the art performance in theory and in practice. Motivated by this result, the thesis goes on with the design of variants of the MDA for different variants of the MOSP problem. For the One-to-One MOSP problem the Targeted MDA (T-MDA) has the same asymptotic running time than the MDA but trades in memory for speed in practice. The additional paths stored during the T-MDA are managed in a pseudo-lazy way. This is a novel way to organize explored paths that ensures that the algorithm’s priority queue stores at most one path per node in the input graph simultaneously. The paths that are held back from the queue are organized in lists that are kept sorted by just prepending or appending paths to them, i.e., using constant time insertions. The resulting implementation of the T-MDA solves bigger instances than the MDA and is also faster. We study the generalization of the Time-Dependent Shortest Path problem to the multiobjective case. We provide a detailed analysis of the generalization’s limitations and discuss when the MDA is applicable. For MOSP instances with large sets of optimal paths, good approximation algorithms are important. We combine the MDA with an outcome space partition technique from the literature to obtain a new FPTAS for the MOSP problem. The resulting MD-FPTAS works also for multiobjective instances of the Time-Dependent Shortest Path problem, which is a novelty. Finally, we use the MDA and its biobjective version, the BDA, to solve the Multiobjective Minimum Spanning Tree (MO-MST) problem and the k-Shortest Simple Path (k-SSP) problem, respectively. For the solution of instances of these two problems, the MDA and the BDA are used as subroutines. An MO-MST instance is solved applying the MDA on a so called transition graph. In this graph paths have equivalent costs to trees in the original graph and thus, the optimal solutions computed by the MDA in the transition graph correspond to optimal trees in the original graph. Since the transition graph has an exponential size w.r.t. the size of the original graph, we discuss new pruning techniques to reduce its number of arcs effectively. The solution of the k-SSP, which is a scalar optimization problem, using a biobjective subroutine is surprising but our new algorithm is state of the art in theory and in practice.In dieser Arbeit beschäftigen wir uns hauptsächlich mit dem Multikriterielle Kürzeste Wege (MOSP) Problem. Es ist eine Verallgemeinerung des klassischen Kürzeste Wege Problems, bei der Kanten im Eingangsgraphen mit d-dimensionalen Vektoren anstelle von Skalaren gewichtet sind. Der Hauptbeitrag der Arbeit ist der Multiobjective Dijkstra Algorithmus (MDA), ein label-setting Algorithmus, der in der Theorie und in der Praxis eine Verbesserung der in der Literatur vorhandenen Ergebnisse darstellt. Motiviert durch dieses Ergebnis entwickeln wir im weiteren Verlauf der Arbeit Varianten des MDAs für verschiedene Varianten des MOSP Problems. Für das Punkt-zu-Punkt MOSP Problem hat der Targeted MDA (T-MDA) die gleiche asymptotische Laufzeit wie der MDA. Durch einen erhöhten Speicherverbrauch ist er in der Praxis aber deutlich schneller. Die zusätzlich gespeicherten Pfade werden auf eine pseudo-lazy Art verwaltet. Dies ist eine neuartige Methode zur Organisation von explorierten Pfaden, die sicherstellt, dass die Größe der Prioritätswarteschlange des Algorithmus auf höchstens einen Pfad pro Knoten im Graphen beschränkt bleibt. Die Pfade, die aus der Warteschlange zurückgehalten werden, werden in Listen organisiert, die durch Voranstellen oder Anhängen von Pfaden sortiert bleiben. Das heißt, dass nur Konstantzeit-Operationen benötigt werden. Der T-MDA löst größere Instanzen als der MDA und ist auch schneller. Wir untersuchen die Verallgemeinerung des zeitabhängigen Kürzeste Wege Problems auf den multikriteriellen Fall. Wir bieten eine detaillierte Analyse der Grenzen und Möglichkeiten dieser Verallgemeinerung und diskutieren, wann der MDA hierauf anwendbar ist. Für MOSP Instanzen mit großen Mengen optimaler Pfade sind gute Approximationsalgorithmen wichtig. Wir kombinieren den MDA mit einer Technik zur Partitionierung des Ergebnisraums aus der Literatur, um einen neuen FPTAS für das MOSP Problem zu erhalten. Der resultierende MD-FPTAS funktioniert auch für Instanzen mit verallgemeinert zeitabhängigen Kostenfunktionen, was neuartig ist. Schließlich verwenden wir den MDA und seine bikriterielle Version als Unterroutinen, um jeweils das Multiobjective Minimum Spanning Tree (MOMST) Problem und das k-Shortest Simple Path (k-SSP) Problem zu lösen. Eine MO-MST Instanz wird gelöst, indem der MDA auf einen sogenannten Übergangsgraphen angewendet wird. In diesem Graphen haben Pfade äquivalente Kosten zu den Bäumen im ursprünglichen Graphen und somit entsprechen die optimalen Lösungen, die vom MDA im Übergangsgraphen berechnet werden, den optimalen Bäumen im ursprünglichen Graphen. Da der Übergangsgraph eine exponentielle Größe im Verhältnis zur Größe des ursprünglichen Graphen hat, diskutieren wir neue Techniken zur Reduzierung seiner Anzahl von Kanten. Die Lösung des k-SSP Problems, ein skalares Optimierungsproblem, unter Verwendung einer bikriteriellen Unterroutine ist überraschend, aber unser neuer Algorithmus ist sowohl in der Theorie als auch in der Praxis auf dem neuesten Stand

Approximation in Multiobjective Optimization with Applications

Over the last couple of decades, the field of multiobjective optimization has received much attention in solving real-life optimization problems in science, engineering, economics and other fields where optimal decisions need to be made in the presence of trade-offs between two or more conflicting objective functions. The conflicting nature of objective functions implies a solution set for a multiobjective optimization problem. Obtaining this set is difficult for many reasons, and a variety of approaches for approximating it either partially or entirely have been proposed. In response to the growing interest in approximation, this research investigates developing a theory and methodology for representing and approximating solution sets of multiobjective optimization problems. The concept of the tolerance function is proposed as a tool for modeling representation quality. Two types of subsets of the set being represented, covers and approximations, are defined, and their properties are examined. In addition, approximating the solution set of the multiobjective set covering problem (MOSCP), one of the challenging combinatorial optimization problems that has seen limited study, is investigated. Two algorithms are proposed for approximating the solution set of the MOSCP, and their approximation quality is derived. A heuristic algorithm is also proposed to approximate the solution set of the MOSCP. The performance of each algorithm is evaluated using test problems. Since the MOSCP has many real-life applications, and in particular designing reserve systems for ecological species is a common field for its applications, two optimization models are proposed in this dissertation for preserving reserve sites for species and their natural habitats

A Polyhedral Study of Mixed 0-1 Set

We consider a variant of the well-known single node fixed charge network flow set with constant capacities. This set arises from the relaxation of more general mixed integer sets such as lot-sizing problems with multiple suppliers. We provide a complete polyhedral characterization of the convex hull of the given set

An Analysis of Some Algorithms and Heuristics for Multiobjective Graph Search

Muchos problemas reales requieren examinar un número exponencial de alternativas para encontrar la elección óptima. A este tipo de problemas se les llama de optimización combinatoria. Además, en problemas reales normalmente se evalúan múltiples magnitudes que presentan conflicto entre ellas. Cuando se optimizan múltiples obje-tivos simultáneamente, generalmente no existe un valor óptimo que satisfaga al mismo tiempo los requisitos para todos los criterios. Solucionar estos problemas combinatorios multiobjetivo deriva comúnmente en un gran conjunto de soluciones Pareto-óptimas, que definen los balances óptimos entre los objetivos considerados. En esta tesis se considera uno de los problemas multiobjetivo más recurrentes: la búsqueda de caminos más cortos en un grafo, teniendo en cuenta múltiples objetivos al mismo tiempo. Se pueden señalar muchas aplicaciones prácticas de la búsqueda multiobjetivo en diferentes dominios: enrutamiento en redes multimedia (Clímaco et al., 2003), programación de satélites (Gabrel & Vanderpooten, 2002), problemas de transporte (Pallottino & Scutellà, 1998), enrutamiento en redes de ferrocarril (Müller-Hannemann & Weihe, 2006), planificación de rutas en redes de carreteras (Jozefowiez et al., 2008), vigilancia con robots (delle Fave et al., 2009) o planificación independiente del dominio (Refanidis & Vlahavas, 2003). La planificación de rutas multiobjetivo sobre mapas de carretera realistas ha sido considerada como un escenario de aplicación potencial para los algoritmos y heurísticos multiobjetivo considerados en esta tesis. El transporte de materias peligrosas (Erkut et al., 2007), otro problema de enrutamiento multiobjetivo relacionado, ha sido también considerado como un escenario de aplicación potencial interesante. Los métodos de optimización de un solo criterio son bien conocidos y han sido ampliamente estudiados. La Búsqueda Heurística permite la reducción de los requisitos de espacio y tiempo de estos métodos, explotando el uso de estimaciones de la distancia real al objetivo. Los problemas multiobjetivo son bastante más complejos que sus equivalentes de un solo objetivo y requieren métodos específicos. Éstos, van desde técnicas de solución exactas a otras aproximadas, que incluyen los métodos metaheurísticos aproximados comúnmente encontrados en la literatura. Esta tesis se ocupa de algoritmos exactos primero-el-mejor y, en particular, del uso de información heurística para mejorar su rendimiento. Esta tesis contribuye análisis tanto formales como empíricos de algoritmos y heurísticos para búsqueda multiobjetivo. La caracterización formal de estos algoritmos es importante para el campo. Sin embargo, la evaluación empírica es también de gran importancia para la aplicación real de estos métodos. Se han utilizado diversas clases de problemas bien conocidos para probar su rendimiento, incluyendo escenarios realistas como los descritos más arriba. Los resultados de esta tesis proporcionan una mejor comprensión de qué métodos de los disponibles sonmejores en situaciones prácticas. Se presentan explicaciones formales y empíricas acerca de su comportamiento. Se muestra que la búsqueda heurística reduce considerablemente los requisitos de espacio y tiempo en la mayoría de las ocasiones. En particular, se presentan los primeros resultados sistemáticos mostrando las ventajas de la aplicación de heurísticos multiobjetivo precalculados. Esta tesis también aporta un método mejorado para el precálculo de los heurísticos, y explora la conveniencia de heurísticos precalculados más informados.Many real problems require the examination of an exponential number of alternatives in order to find the best choice. They are the so-called combinatorial optimization problems. Besides, real problems usually involve the consideration of several conflicting magnitudes. When multiple objectives must be simultaneously optimized, there is generally not an optimal value satisfying the requirements for all the criteria at the same time. Solving these multiobjective combinatorial problems commonly results in a large set of Pareto-optimal solutions, which define the optimal tradeoffs between the objectives under consideration. One of most recurrent multiobjective problems is considered in this thesis: the search for shortest paths in a graph, taking into account several objectives at the same time. Many practical applications of multiobjective search in different domains can be pointed out: routing in multimedia networks (Clímaco et al., 2003), satellite scheduling (Gabrel & Vanderpooten, 2002), transportation problems (Pallottino & Scutellà, 1998), routing in railway networks (Müller-Hannemann & Weihe, 2006), route planning in road maps (Jozefowiez et al., 2008), robot surveillance (delle Fave et al., 2009) or domain independent planning (Refanidis & Vlahavas, 2003). Multiobjective route planning over realistic road maps has been considered as a potential application scenario for the multiobjective algorithms and heuristics considered in this thesis. Hazardous material transportation (Erkut et al., 2007), another related multiobjective routing problem, has also been considered as an interesting potential application scenario. Single criterion shortest path methods are well known and have been widely studied. Heuristic Search allows the reduction of the space and time requirements of these methods, exploiting estimates of the actual distance to the goal. Multiobjective problems are much more complex than their single-objective counterparts, and require specific methods. These range from exact solution techniques to approximate ones, including the metaheuristic approximate methods usually found in the literature. This thesis is concerned with exact best-first algorithms, and particularly, with the use of heuristic information to improve their performance. This thesis contributes both formal and empirical analysis of algorithms and heuristics for multiobjective search. The formal characterization of algorithms is important for the field. However, empirical evaluation is also of great importance for the real application of these methods. Several well known classes of problems have been used to test their performance, including some realistic scenarios as described above. The results of this thesis provide a better understanding of which of the available methods are better in practical situations. Formal and empirical explanations of their behaviour are presented. Heuristic search is shown to reduce considerably space and time requirements in most situations. In particular, the first systematic results showing the advantages of the application of precalculated multiobjective heuristics are presented. The thesis also contributes an improved method for heuristic precalculation, and explores the convenience of more informed precalculated heuristics.This work is partially funded by / Este trabajo está financiado por: Consejería de Economía, Innovación, Ciencia y Empresa. Junta de Andalucía (España) Referencia: P07-TIC-0301