Semidefinite Programming. Methods and algorithms for energy management Agnès Maher To cite this version: Agnès Maher. Semidefinite Programming. Methods and algorithms for energy management. Other [cs.OH]. Université Paris Sud - Paris XI, 2013. English. NNT : 2013PA112185. tel-00881025 HAL Id: tel-00881025 https://tel.archives-ouvertes.fr/tel-00881025 Submitted on 7 Nov 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. UNIVERSITE PARIS-SUD ECOLE DOCTORALE INFORMATIQUE DE PARIS-SUD (ED 427) Laboratoire de Recherche en Informatique (LRI) DISCIPLINE INFORMATIQUE THESE DE DOCTORAT soutenue le 26/09/2013 par Agnès Gorge Semidefinite Programming : methods and algorithms for energy management Directeur de thèse : Abdel LISSER Professeur (LRI - Université Paris-Sud) Composition du jury : Président du jury : Rapporteurs : Franz RENDL Professeur (Université Alpen-Adria, Klagenfurt) Didier HENRION Directeur de recherche (LAAS CNRS) Examinateurs : Alain DENISE Professeur (LRI - Université Paris-Sud) Abdel LISSER Professeur (LRI - Université Paris-Sud) Abdelatif MANSOURI Professeur (Université Cadi Ayyad, Marrakech) Michel MINOUX Professeur (LIP6 - UPMC) Riadh ZORGATI Docteur (EDF R & D) Abstract This thesis aims at exploring the potentialities of a powerful optimization technique, namely Semidefinite Programming, for addressing some difficult problems of energy management. This relatively young area of convex and conic optimization has undergone a rapid development in the last decades, partly thanks to the design of efficient algorithms for their resolution, and because numerous NP-hard problems can be approached using semidefinite programming. In the present thesis, we pursue two main objectives. The first one consists of exploring the potentiality of semidefinite programming to provide tight relaxations of combinatorial and quadratic problems. This line of research was motivated both by the promising results obtained in this direction [108, 186, 245] and by the combinatorial and quadratic features presented by energy management problems. The second one deals with the treatment of uncertainty, an issue that is also of paramount importance in energy management problems. Indeed, due to its versatility, SDP is well-known for providing numerous possibilities of dealing with uncertainty. In particular, it offers a way of modelling the deterministic counterpart of robust optimization problems, or more originally, of distributionnally robust optimization problems. The first part of this thesis contains the theoretical results related to SDP, starting by the underlying theory of convex and conic optimization, followed by a focus on semidefinite programming and its most famous applications. In particular, we provide a comprehensive and unified framework of the different methods proposed in the literature to design SDP relaxations of QCQP The second part is composed of the last three chapters and presents the application of SDP to energy management. Chapter 4 provides an introduction to energy management problems, with a special emphasis on one of the most challenging energy management problem, namely the Nuclear Outages Scheduling Problems. This problem was selected both for being a hard combinatorial problem and for requiring the consideration of uncertainty. We present at the end of this chapter the different models that we elaborated for this problem. The next chapter reports the work related to the first objective of the thesis, i.e., the design of semidefinite programming relaxations of combinatorial and quadratic programs. On certain problems, these relaxations are provably tight, but generally it is desirable to reinforce them, by means of tailor- made tools or in a systematic fashion. We apply this paradigm to different models of the Nuclear Outages Scheduling Problem. Firstly, we consider a complete model that takes the form of a MIQP. We apply the semidefinite relaxation and reinforce it by addition of appropriate constraints. Then, we take a step further by developing a method that automatically generates such constraints, called cutting planes. For the design of this method, we start by providing a framework for unification of many seemingly disparate cutting planes that are proposed in the literature by noting that all these constraints are linear combinations of the initial quadratic constraints of the problem and of the pair-wise product of the linear constraints of the problem (including bounds constraints). Subsequently, we focus on specific part of the problem, namely the maximal lapping constraint, which takes the form of aT x / [b,c], where x are binary variables. This constraint presents modelling difficulty due to its disjunctive∈ nature. We aim at comparing three possible modelisations and for each of them, computing different relaxations based on semidefinite programming and linear programming. Finally, we conclude this chapter by an experiment of the Lasserre’s hierarchy, a very powerful tool dedicated to polynomial optimization that builds a sequence of semidefinite relaxations whose optimal values tends to the optimal value of the considered problem. Thus, we fulfilled the first objective of this thesis, namely exploring the potentiality of semidef- inite programming to provide tight relaxations of combinatorial and quadratic problems. The second objective is to examine how semidefinite programming can be used to tackle uncertainty. To this end, three different works are carried out. First, we investigate a version of the nuclear outages scheduling problem where uncertainty is described in the form of equiprobable scenarios and the constraints in- volving uncertain parameters have to be satisfied up to a given level of probability. It is well-known that this model admits a deterministic formulation by adding binary variables. Then the obtained problem is a combinatorial problem and we apply semidefinite programming to compute tight bounds of the optimal value. We have also implemented a more original way of dealing with uncertainty, which admits a deterministic counterpart, or a conservative approximation, under the form of a semidefinite program. This method, that has received much attention recently, is called distributionnally robust optimization and can be seen as a compromise between stochastic optimization, where the probability distribution is required, and robust optimization, where only the support is required. Indeed, in distributionnally robust optimization, the support and some moments of the probability distribution are required. In our case, we assume that the support, the expected value and the covariance are known and we compare the benefits of this method w.r.t other existing approaches, based on Second-Order Cone Program, that rely on the application of the Boole’s inequality, to convert the joint constraint into individual ones, combined to the Hoeffding’s inequality, in order to get a tractable conservative approximation of the chance constraints. Finally, we carried out a last experiment that combines both uncertainty and combinatorial aspects. Indeed, many deterministic counterpart or conservative approximation of Linear Program (LP) subject to uncertainty give rise to a Second-Order Cone Program (SOCP). In the case of a Mixed- Integer Linear Program, we obtain a MISOCP, for which there is no reference resolution method. Then we investigate the strength of the SDP relaxation for such problems. Central to our approach is the reformulation as a non convex Quadratically Constrained Quadratic Program (QCQP), which brings us in the framework of binary quadratically constrained quadratic program. This allows to apply the well-known semidefinite relaxation for such problems. When necessary, this relaxation is tightened by adding constraints of the initial problem. We report promising computational results indicating that the semidefinite relaxation improves significantly the continuous relaxation (112% on average) and often provides a lower bound very close to the optimal value. In addition, computational time for obtaining these results remains reasonable. In conclusion, despite practical difficulties mainly due to the fact that SDP is not a mature technology yet, it is nonetheless a very promising optimization method, that combines all the strengths of conic programming and offers great opportunities for innovation at EDF R&D, both in energy management and engineering or financial issues. Résumé Cette thèse se propose d’explorer les potentialités qu’offre une méthode prometteuse de l’op- timisation convexe et conique, la programmation semi-définie positive (SDP), pour les problèmes de management d’énergie. La programmation semi-définie positive est en effet l’une des méthodes ayant le plus attiré l’atten- tion de la communauté scientifique ces dernières années, du fait d’une part de la possibilité de pouvoir résoudre ses instances en temps polynomial grâce à des solveurs performants. D’autre part, il s’est avéré que de nombreux problèmes d’optimisation NP-difficiles peuvent être approximés au moyen d’un SDP. Ce rapport débute par un résumé des principaux