Disjunctive Inequalities: Applications and Extensions

A general optimization problem can be expressed in the form min{cx: x ∈ S}, (1) where x ∈ R n is the vector of decision variables, c ∈ R n is a linear objective function and S ⊂ R n is the set of feasible solutions of (1). Because S is generall

Closing the Gap in Linear Bilevel Optimization: A New Valid Primal-Dual Inequality

International audienceLinear bilevel optimization problems are often tackled by replacing the linear lower-level problem with its Karush-Kuhn-Tucker (KKT) conditions. The resulting single-level problem can be solved in a branch-and-bound fashion by branching on the complementarity constraints of the lower-level problem's optimality conditions. While in mixed-integer single-level optimization branch-and-cut has proven to be a powerful extension of branch-and-bound, in linear bilevel optimization not too many bilevel-tailored valid inequalities exist. In this paper, we briefly review existing cuts for linear bilevel problems and introduce a new valid inequality that exploits the strong duality condition of the lower level. We further discuss strengthened variants of the inequality that can be derived from McCormick envelopes. In a computational study, we show that the new valid inequalities can help to close the optimality gap very effectively on a large test set of linear bilevel instances

A solution to bi/tri-level programming problems using particle swarm optimization

© 2016 Elsevier Inc. Multilevel (including bi-level and tri-level) programming aims to solve decentralized decision-making problems that feature interactive decision entities distributed throughout a hierarchical organization. Since the multilevel programming problem is strongly NP-hard and traditional exact algorithmic approaches lack efficiency, heuristics-based particle swarm optimization (PSO) algorithms have been used to generate an alternative for solving such problems. However, the existing PSO algorithms are limited to solving linear or small-scale bi-level programming problems. This paper first develops a novel bi-level PSO algorithm to solve general bi-level programs involving nonlinear and large-scale problems. It then proposes a tri-level PSO algorithm for handling tri-level programming problems that are more challenging than bi-level programs and have not been well solved by existing algorithms. For the sake of exploring the algorithms' performance, the proposed bi/tri-level PSO algorithms are applied to solve 62 benchmark problems and 810 large-scale problems which are randomly constructed. The computational results and comparison with other algorithms clearly illustrate the effectiveness of the proposed PSO algorithms in solving bi-level and tri-level programming problems

A New General-Purpose Algorithm for Mixed-Integer Bilevel Linear Programs

Bilevel optimization problems are very challenging optimization models arising in many important practical contexts, including pricing mechanisms in the energy sector, airline and telecommunication industry, transportation networks, critical infrastructure defense, and machine learning. In this paper, we consider bilevel programs with continuous and discrete variables at both levels, with linear objectives and constraints (continuous upper level variables, if any, must not appear in the lower level problem). We propose a general-purpose branch-and-cut exact solution method based on several new classes of valid inequalities, which also exploits a very effective bilevel-specific preprocessing procedure. An extensive computational study is presented to evaluate the performance of various solution methods on a common testbed of more than 800 instances from the literature and 60 randomly generated instances. Our new algorithm consistently outperforms (often by a large margin) alternative state-of-the-art methods from the literature, including methods exploiting problem-specific information for special instance classes. In particular, it solves to optimality more than 300 previously unsolved instances from the literature. To foster research on this challenging topic, our solver is made publicly available online

A New General-Purpose Algorithm for Mixed-Integer Bilevel Linear Programs

International audienceBilevel optimization problems are very challenging optimization models arising in many important practical contexts, including pricing mechanisms in the energy sector, airline and telecommunication industry, transportation networks, critical infrastructure defense, and machine learning. In this paper, we consider bilevel programs with continuous and discrete variables at both levels, with linear objectives and constraints (continuous upper level variables, if any, must not appear in the lower level problem). We propose a general-purpose branch-and-cut exact solution method based on several new classes of valid inequalities, which also exploits a very effective bilevel-specific preprocessing procedure. An extensive computational study is presented to evaluate the performance of various solution methods on a common testbed of more than 800 instances from the literature and 60 randomly generated instances. Our new algorithm consistently outperforms (often by a large margin) alternative state-of-the-art methods from the literature, including methods exploiting problem-specific information for special instance classes. In particular, it solves to optimality more than 300 previously unsolved instances from the literature. To foster research on this challenging topic, our solver is made publicly available online

Three essays on bilevel optimization algorithms and applications

This thesis consists of three journal papers I have worked on during the past three years of my PhD studies. In the first paper, we presented a multi-objective integer programming model for the gene stacking problem. Although the gene stacking problem is proved to be NP-hard, we have been able to obtain Pareto frontiers for smaller sized instances within one minute using the state-of-the-art commercial computer solvers in our computational experiments. In the second paper, we presented an exact algorithm for the bilevel mixed integer linear programming (BMILP) problem under three simplifying assumptions. Compared to these existing ones, our new algorithm relies on weaker assumptions, explicitly considers infinite optimal, infeasible, and unbounded cases, and is proved to terminate infinitely with the correct output. We report results of our computational experiments on a small library of BMILP test instances, which we have created and made publicly available online. In the third paper, we presented the watermelon algorithm for the bilevel integer linear programming (BILP) problem. To our best knowledge, it is the first exact algorithm which promises to solve all possible BILPs, including finitely optimal, infeasible, and unbounded cases. What is more, our algorithm does not rely on any simplifying condition, allowing even the case of unboundedness for the high point problem. We prove that the watermelon algorithm must finitely terminate with the correct output. Computational experiments are also reported showing the efficiency of our algorithm

Multilevel decision-making: A survey

© 2016 Elsevier Inc. All rights reserved. Multilevel decision-making techniques aim to deal with decentralized management problems that feature interactive decision entities distributed throughout a multiple level hierarchy. Significant efforts have been devoted to understanding the fundamental concepts and developing diverse solution algorithms associated with multilevel decision-making by researchers in areas of both mathematics/computer science and business areas. Researchers have emphasized the importance of developing a range of multilevel decision-making techniques to handle a wide variety of management and optimization problems in real-world applications, and have successfully gained experience in this area. It is thus vital that a high quality, instructive review of current trends should be conducted, not only of the theoretical research results but also the practical developments in multilevel decision-making in business. This paper systematically reviews up-to-date multilevel decision-making techniques and clusters related technique developments into four main categories: bi-level decision-making (including multi-objective and multi-follower situations), tri-level decision-making, fuzzy multilevel decision-making, and the applications of these techniques in different domains. By providing state-of-the-art knowledge, this survey will directly support researchers and practical professionals in their understanding of developments in theoretical research results and applications in relation to multilevel decision-making techniques

Analysis of investment decision making in power systems under environmental regulations and uncertainties

The dissertation focuses on the study of environmental policies and their impacts on the power systems\u27 planning. It consists of three parts, each of which addresses a single problem on environmental policies and generation expansion planning. In the first part of the dissertation, I compared the cap-and-trade policy and various carbon tax policies in a single period under the generation expansion framework. The problem was modeled as a bilevel problem where the lower level competing generation companies maximized their own profits under the regulations of the upper level. The policies were compared via their effectiveness and efficiency. Effectiveness referred to a policy\u27s capability to control the amount of carbon emissions, and efficiency was measured with respect to five criteria: emissions price, renewable energy portfolio, total generation, total profit of generation companies and grid owner, and government revenue. In the second part, the model was extended to multi-period planning to gain better views into market dynamics. Cap-and-trade and four variations of carbon tax policies were integrated in a game-theory based generation expansion planning model to assess their impacts on new investments in renewable energy generation capacity. The most efficient tax policy and variations were obtained using inverse equilibrium models. The third part complemented the previous parts by conducting a realistic case study on the generation expansion planning under uncertainty. It studied the formulation and solution of investment decisions in new generation under the explicit representation of environmental policies and their associated uncertainty. A three-layer framework was proposed to study the investment decisions. The operations layer was used to represent the transmission physical flows under economic dispatch in the network; the assessment layer completed comprehensive assessment of candidate investment plans under uncertainty; the optimization layer was designed to compare the optimal investment decisions for the decision makers based on the optimization criteria. Our framework was tested on a realistic 240-bus WECC network, taking into account representative scenarios and investment plans

Semidefinite programming approaches and software tools for quadratic programs with linear complementarity constraints

RÉSUMÉ : Dans le domaine de la théorie des jeux, il est intéressant de créer un équilibre dynamique entre les agents afin qu’ils s’influencent de façon asymétrique. Le meneur affecte les règles du jeu, mais les choix subséquents du suiveur affectent la valeur de l’objectif du meneur. La dynamique meneur-suiveur est un outil puissant permettant de décrire un grand nombre de scénarios de jeux dans un contexte réel. Toutefois, les problèmes d’équilibre demeurent difficiles en pratique sauf pour quelques types de problèmes largement étudiés en théorie tel le problème linéaire bi-niveau. Cette thèse tente de déterminer si les relaxations sous la forme de problèmes semi-définis, problèmes quadratiques avec contraintes de complémentarité linéaires sont efficaces. Cette classe de problème est équivalente aux problèmes d’équilibre. Une fonction objectif quadratique est particulièrement intéressante car la littérature dans ce domaine n’est pas complète et les relaxations semi-définies sont souvent efficaces pour les problèmes avec des fonctions objectif et/ou des contraintes quadratiques non-convexes. Nous présentons une relaxation de base qui n’est pas coûteuse en temps de calcul puis nous discutons d’un grand nombre de contraintes qui permettent de resserrer la relaxation de façon significative. L’évaluation de l’efficacité de la relaxation, lorsque toutes ces contraintes sont utilisées, montre que cela mène à des difficultés d’implémentation numériques pour le solveur de points intérieurs qui résout le problème semi-défini. Nous discutons des raisons expliquant cela puis nous utilisons une autre approche afin d’éliminer cette difficulté. Cet algorithme démarre avec la relaxation de base renforcée avec une seule contrainte d’égalité agrégée puis ajoute de façon itérative des coupes resserrant la relaxation. Éventuellement, la relaxation est renforcée à son maximum alors que seule une fraction des coupes a été ajoutée. Les résultats numériques montrent que cette approche ne permet pas d’améliorer les bornes du problème semi-défini lorsque des coupes sont ajoutées. Ce n’est pas une faiblesse de la méthode mais cela démontre que le modèle de base est déjà une relaxation forte. Ainsi, l’aggrégation des contraintes en une seule contrainte d’égalité est très efficace pour renforcer la relaxation et ajoute peu de difficulté à son implémentation en pratique. Des recommandations sont émises concernant le choix des paramètres pour la méthode d’ajout de coupes de façon itérative. Les relaxations semi-définies sont surtout utilisées pour borner les problèmes quadratiques difficiles. Les relaxations SDP des problèmes QPLCC peuvent être utilisées de cette façon, pour borner les noeuds des arbres branch and bound, mais nous sommes intéressés à utiliser toute l’information contenue dans la matrice de solution X∗ du problème SDP. Lorsque cette matrice est de rang 1, elle peut être utilisée pour retrouver la solution globale dans l’espace d’état du problème original non-relaxé. Nous définissons un point candidat comme étant un point estimant la solution globale d’un problème et nous présentons 4 façons de retrouver une solution dans l’espace d’état original pour une matrice X∗ de rang arbitraire, X∗ étant la solution de la relaxation SDP. Ce point candidat n’est pas spécifique aux problèmes QPLCC et pourrait être appliqué à d’autres problèmes. Des résultats numériques sont effectués afin de montrer que les points candidats sont des estimateurs de la solution globale. Nous présentons aussi des procédures afin d’utiliser les capacités de "warmstart" des solveurs en utilisant ce point candidat et démontrons leur impact. En plus de contribuer à l’avancement des connaissances des problèmes QPLCC, nous avons aussi contribué à la communauté de recherche des logiciels traitant ces problèmes. Nous avons choisi Python comme langage de programmation puisque plusieurs librairies sont disponibles pour l’optimisation convexe, mais aussi pour sa capacité à interagir avec des solveurs externes codés dans d’autres langages de programmation. Nous avons créé des outils pour les problèmes QPLCC, par exemple en les formulant en langage AMPL et GAMS, nous avons résolu les QPLCC et/ou les problèmes SDP en utilisant des solveurs Pytyon, des solveurs installés localement ou la librairie NEOS. Tous les résultats numériques présentés dans cette thèse ont été effectués avec les librairies présentées dans cette thèse. Nous espérons que d’autres chercheurs dans le domaine QLPCC utiliserons nos librairies pour construire leurs propres méthodes de résolution et pour simplifier les comparaisons avec d’autres solveurs.----------ABSTRACT : The leader follower dynamic seen in bilevel programming and equilibrium problems has the potential to unlock new doors in economic modeling and enable the realistic modeling of many problems of keen interest. The leader’s choices affect the rules of the game which is played by the follower, but the follower’s subsequent choices also impact the objective value achieved by the leader. Conceptually, the leader-follower dynamic is a valuable tool for describing any number of competitive real-world scenarios. However, to date equilibrium problems remain difficult in practice except for a handful of well studied problem classes such as the linear linear bilevel program. This thesis is concerned with how effective semidefinite programming (SDP) relaxations can be constructed for quadratic programs with linear complementarity constraints (QPLCCs), a problem class which can equivalently model a class of equilibrium problems. The case of a general quadratic objective function is of particular interest since the literature in this area has not yet reached full maturity, and since semidefinite programming relaxations have often been effective for problems with nonconvex quadratic objective functions and constraints. We present a base relaxation which is relatively computationally inexpensive, and then we present and discuss a number of tightening constraints which can have a dramatic tightening effect. How- ever, in evaluating the effectiveness of the relaxation when all such constraints are used, we observe that blindly imposing all tightening constraints of the proposed types often leads to numerical difficulties for the interior point solver solving the semidefinite program. We discuss a possible reason for this, and finally we counteract it by developing an algorithm which begins with a middle ground model (the base relaxation strengthened with a single aggregated equality constraint) and iteratively adds tightening constraints to eventually obtain the tightness of the full relaxation while using a small fraction of the constraint pool in practice. In testing the iterative method with the middle ground model, we find that the SDP bound often doesn’t improve over the course of the iterative method. This is not a flaw of the cut finding procedure, but instead demonstrates that the middle ground model is already as tight as the fully constrained model for these problems, establishing the aggregated equality constraint as an extremely effective strengthening measure which adds very little additional computational difficulty to the problem. Recommendations are made regarding parameter choice for the iterative method. Semidefinite relaxations are most commonly used to bound difficult quadratic problems. SDP relaxations of QPLCCs can certainly be used this way, to bound nodes of a branch and bound tree, but we are also interested in using the information contained in the SDP solution matrix X∗ to full advantage. SDP relaxations are commonly designed so that an SDP solution X∗ which is rank one can be mapped back to the space of the unrelaxed problem to give a global solution. We define the notion of a candidate point as a point which is intended to estimate a problem’s global solution, and we present four ways a candidate point for an SDP relaxation solution X∗ of arbitrary rank can be mapped back to the space of the original problem. The candidate point concept and definitions are not specific to the QPLCC and could be applied to other problems. We perform computational tests to support discussion of the different candidate points’ suitability as an estimate of the problem’s global solution. We also present procedures for assisting local or global solvers by warmstarting with the candidate point, and demonstrate the impact in both cases. In addition to contributing to the state of knowledge for QPLCCs, it has also been our goal to contribute software to the research community working on these problems. We have chosen Python as our development language based on the existence of a number of good packages for numerical work generally and convex optimization specifically, and also based on its abilities to act as glue between other services such as external solvers in other languages. We have made tools for modeling QPLCCs, exporting them for other languages (AMPL, GAMS), formulating SDP relaxations, and solving QPLCCs and/or SDP problems using native Python solvers, locally installed languages/solvers, or the NEOS public server. All the computational work presented in this thesis has been executed using the packages presented in this paper. It is our hope that other researchers in the field of QPLCCs will use our packages to build their own solution methods and to simplify the process of testing against various solvers