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Solving Well-Structured MINLP Problems UNIVERSITE´ PARIS 13 LABORATOIRE D’INFORMATIQUE DE PARIS NORD HABILITATION A` DIRIGER DES RECHERCHES specialit´ e´ SCIENCES (Informatique) par Claudia D’Ambrosio Solving well-structured MINLP problems Jury: Emilio Carrizosa Universidad de Sevilla Rapporteur Sourour Elloumi Ecole´ Nationale Superieure´ de Examinateur Techniques Avancees´ Ignacio Grossmann Carnegie Mellon University Rapporteur Jean-Bernard Lasserre CNRS-LAAS Examinateur Ruth Misener Imperial College London Rapporteur Fred´ eric´ Roupin Universite´ Paris 13 Parrain Roberto Wolfler Calvo Universite´ Paris 13 Parrain Contents Acknowledgments iii Keywordsv Preface vii I Curriculum Vitae1 II Research Summary 23 1 Introduction to MINLP Problems and Methods 25 1.1 Modeling problems by Mathematical Programming............. 26 1.1.1 Reformulations............................ 26 1.1.2 Relaxations.............................. 27 1.1.3 Approximations........................... 27 1.1.4 Solution methods........................... 28 1.2 Mixed Integer Linear Programming Technology............... 29 1.3 Non Linear Programming Technology.................... 30 1.4 Mixed Integer Non Linear Programming Technology............ 30 2 Exact Methods for MINLP problems 35 2.1 Methods based on strengthening of classic relaxations............ 35 2.1.1 Euclidean Steiner Tree........................ 36 2.1.2 The pooling problem and its variant with binary variables..... 38 2.2 On Separable Functions............................ 41 2.2.1 SC-MINLP.............................. 42 2.2.2 Polyopt................................ 45 3 Heuristic Methods 47 3.1 General methods............................... 47 47subsection.3.1.1 i ii CONTENTS 3.1.2 Optimistic approximation...................... 49 3.2 Tailored methods............................... 51 3.2.1 Non linear Knapsack Problem.................... 51 4 Other contributions 53 4.1 Feasibility issues on Hydro Unit Commitment................ 53 4.2 MILP models for the selection of a small set of well-distributed points... 54 4.3 Well placement in reservoir engineering................... 54 4.4 A heuristic for multiobjective convex MINLPs................ 55 4.5 Bilevel programming and smart grids.................... 55 4.6 Feasibility Pump for aircraft deconfliction problems............. 56 4.7 On the Product Knapsack Problem...................... 56 4.8 Learning to Solve Hydro Unit Commitment Problems in France...... 57 5 Conclusions and Future Research 59 Bibliography 61 A Selected Publications 67 Acknowledgments The road that brought me here was long. The more I think about when my passion to Optimization started, the more I’m convinced that it dates back to when I was a third- year bachelor student at University of Bologna. One of the optional classes I choose and attended was Practical Tools for Optimization taught by Professor Andrea Lodi: besides a natural inclination for the topic, the enthusiasm of Andrea was so captivating that for the Master I choose all the classes on Optimization that were offered and a Master Thesis on mixed integer non linear optimization, a topic I got immediately addicted to. In 2006 I started my Ph.D. at University of Bologna where the O.R. group welcomed me warmly from the very beginning. I deeply thank all the members of this wonderful group, with a special thought to Andrea who, as Ph.D. advisor, was and is an inspiration to me. Back in 2007 I had the chance to spend my summer working as intern at the IBM T.J. Watson Research Center: Jon Lee and Andreas Wachter¨ were great manager and tutor, respectively, and all the colleagues at IBM helped to make it a great experience. Then in 2010 I got a post-doc position at University of Wisconsin-Madison and I had the chance to be supervised by Jeff Linderoth and Jim Luedtke. I learned a lot in 6 months and the festive environment made me enjoy very much my Mid-West experience. Thanks also to all the students of the group who were lovely officemates. In October 2011 I moved to France to join the CNRS family and, in particular, I was affiliated to the SYSMO team at LIX, Ecole´ Polytechnique. Among the SYSMO mem- bers, I would like to thank, in particular, Leo Liberti who’s a precious colleague and friend, Evelyne Rayssac who helps me a lot since the very first moment to survive to the crazy bu- reaucracy, the former and current Ph.D. student Claire Lizon, Youcef Sahraoui, Luca Men- carelli, Vy Khac Ky, Gustavo Dias, Oliver Wang, Kostas Tavlaridis-Gyparakis, Gabriele Iommazzo, and the former and current post-docs Sonia Toubaline, Pierre-Louis Poirion, and Dimitri Thomopulos. Moreover, I would like to thank Fabio Roda and Sylvie Tonda who worked and works hard to convince industries about the importance of Optimization. Earlier this year the SYSMO team gave way to the DASCIM team and I thank Michalis Vazirgiannis who took the lead of such challenging project. I hope we have several years of fruitful collaborations ahead. I started discussing about this HDR with Roberto Wolfler-Calvo and Frederic Roupin several years ago, maybe 5 years ago. They immediately accepted to be my tutors and iii iv ACKNOWLEDGMENTS were kind enough to wait 5 years to make this happen. Thanks a lot for your help and support, it was really precious to me the fact that you did not put pressure on me to finish this when I needed to focus on something else but you did when I needed to finalize the manuscript. I warmly thank also the other members of the HDR, namely the “rapporteurs” Emilio Carrizosa, Ignacio Grossmann, and Ruth Misener, and the “examinateurs” Sourour Elloumi and Jean Bernard Lasserre. It is really an honor to have them in my HDR jury. I made a huge step forward with the writing of this HDR at the end of 2015 when I spent 3 months in Montreal.´ Andrea Lodi invited me to visit him and gave me the opportunity to work in the dynamic environment of GERAD and CIRRELT. I would like to thank him together with all the GERAD and CIRRELT members, in particular Miguel Anjos and Zhao Sun, fantastic collaborators, and Marta Pascoal, with who I shared the office, lunches, and lovely times. I also would like to thank Sonia Vanier & the equipe´ SAMM of Universite´ Paris 1 who was so kind to host me when I had to “hide” to focus on finalizing this manuscript: the nice desk with a view on Paris, the enjoyable coffee break with plenty of chocolate and laughs, and your hospitality helped me a lot to achieve the end of this manuscript writing. During the last ten years I had the chance to meet and collaborate with several col- leagues, each of which taught me a lot and make me enjoy my work. I thank all the co-authors of the works presented in this manuscript and of all my on-going projects: it is also thanks to you that I am still passionate about research! Last but not least, I need to devote a special thought to my parents, my sister, my friends, and, in particular, to Alessandro who supported me constantly and with a unique patience during the last 9 years. I’m really lucky and grateful for having such a generous and understanding husband. Paris, 2018 Claudia D’Ambrosio Keywords Mixed integer non linear programming Non convex problems Well-structured problems Real-world applications Modeling v vi Keywords Preface This manuscript summarizes the main research topics I have been working on since my Ph.D. at University of Bologna (Italy). The broader common topic is mixed integer non linear programming (MINLP) problems, i.e., optimization problems in which both contin- uous and integer decision variables are present and, in general, non linear functions define the objective to minimize and the constraints to satisfy. In this context, I have been working both on theoretical aspects and real-world applica- tions. All the research performed aims at solving a class of MINLP problems by exploring and exploiting the specific properties that characterize such class. The manuscript is composed of two parts. PartI includes a complete curriculum vitæ and the most relevant publications. PartII is divided as follows: In Chapter1 an introduction on MINLP problems and on classical methods to solve it and its most widely used subclasses is presented. The research concerning exact methods for classes of MINLPs is described in Chap- ter2 . In particular, the chapter is composed of two main parts: i. methods based on the strengthening of classical relaxations and ii. methods based on non standard relaxations. In Chapter3 we discuss heuristic methods for generic MINLP problems, i.e., Feasi- bility Pump and Optimistic Approximation for non convex MINLPs, and for specific problems like the non linear knapsack and its generalizations. The methods for non convex MINLP show special properties for subclasses of MINLPs. For example, the heuristics for the non linear knapsack problem exploit the fact that the only non linear functions, i.e., the profit and weight functions, are separable and non decreasing. Other contributions are summarized in Chapter4 . Finally, conclusions and future research are discussed in Chapter5 . vii viii PREFACE Part I Curriculum Vitae 1 Curriculum Vitae Claudia D’Ambrosio June 11, 2018 CNRS Charg´eede Recherche and Charg´eed’Enseignement at: LIX, Laboratoire d’Informatique de l’Ecole´ Polytechnique 91128 Palaiseau CEDEX, France Office: LIX 1066 (Batiment Alan Turing, first floor) E-mail address: [email protected] Office phone n.: +33.1.77.57.80.06 Personal Information and Education From January 2006 to April 2009: PhD student in “Automatic Control and Operations Research” at the Dipartimento di Elettronica, Informatica e Sis- temistica (DEIS), University of Bologna (Italy). Thesis “Application-oriented Mixed Integer Non-Linear Programming”, advisor Prof. Andrea Lodi. 21st March 2005: Master degree in Computer Science Engineering at the Uni- versity of Bologna (Italy). Thesis: “Algorithms for the Water Network De- sign Problem” advisor Prof. Paolo Toth, co-advisors Prof. Andrea Lodi and Dr.
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