Advances in Polynomial Optimization

Polynomial optimization has a wide range of practical applications in fields such as optimal control, energy and water networks, facility location, management science, and finance. It also generalizes relevant optimization problems thoroughly studied in the literature, such as mixed-binary linear optimization, quadratic optimization, and complementarity problems. As finding globally optimal solutions is an extremely challenging task, the development of efficient techniques for solving polynomial optimization problems is of particular relevance. In this thesis we provide a detailed study of different techniques to solve this kind of problems and we introduce some nobel approaches in this field, including the use of statistical learning techniques. Furthermore, we also present a practical application of polynomial optimization to finance and more specifically, portfolio design

Linear Programming Relaxations of Quadratically Constrained Quadratic Programs

We investigate the use of linear programming tools for solving semidefinite programming relaxations of quadratically constrained quadratic problems. Classes of valid linear inequalities are presented, including sparse PSD cuts, and principal minors PSD cuts. Computational results based on instances from the literature are presented.Comment: Published in IMA Volumes in Mathematics and its Applications, 2012, Volume 15

Of keyboards and beyond - optimization in human-computer interaction

In this thesis, we present optimization frameworks in the area of Human-Computer Interaction. At first, we discuss keyboard layout problems with a special focus on a project we participated in, which aimed at designing the new French keyboard standard. The special nature of this national-scale project and its optimization ingredients are discussed in detail; we specifically highlight our algorithmic contribution to this project. Exploiting the special structure of this design problem, we propose an optimization framework that was efficiently computes keyboard layouts and provides very good optimality guarantees in form of tight lower bounds. The optimized layout that we showed to be nearly optimal was the basis of the new French keyboard standard recently published in the National Assembly in Paris. Moreover, we propose a relaxation for the quadratic assignment problem (a generalization of keyboard layouts) that is based on semidefinite programming. In a branch-and-bound framework, this relaxation achieves competitive results compared to commonly used linear programming relaxations for this problem. Finally, we introduce a modeling language for mixed integer programs that especially focuses on the challenges and features that appear in participatory optimization problems similar to the French keyboard design process.Diese Arbeit behandelt Ansätze zu Optimierungsproblemen im Bereich Human-Computer Interaction. Zuerst diskutieren wir Tastaturbelegungsprobleme mit einem besonderen Fokus auf einem Projekt, an dem wir teilgenommen haben: die Erstellung eines neuen Standards für die französische Tastatur. Wir gehen auf die besondere Struktur dieses Problems und unseren algorithmischen Beitrag ein: ein Algorithmus, der mit Optimierungsmethoden die Struktur dieses speziellen Problems ausnutzt. Mithilfe dieses Algorithmus konnten wir effizient Tastaturbelegungen berechnen und die Qualität dieser Belegungen effektiv (in Form von unteren Schranken) nachweisen. Das finale optimierte Layout, welches mit unserer Methode bewiesenermaßen nahezu optimal ist, diente als Grundlage für den kürzlich in der französischen Nationalversammlung veröffentlichten neuen französischen Tastaturstandard. Darüberhinaus beschreiben wir eine Relaxierung für das quadratische Zuweisungsproblem (eine Verallgemeinerung des Tastaturbelegungsproblems), die auf semidefinieter Programmierung basiert. Wir zeigen, dass unser Algorithmus im Vergleich zu üblich genutzten linearen Relaxierung gut abschneidet. Abschließend definieren und diskutieren wir eine Modellierungssprache für gemischt integrale Programme. Diese Sprache ist speziell auf die besonderen Herausforderungen abgestimmt, die bei interaktiven Optimierungsproblemen auftreten, welche einen ähnlichen Charakter haben wie der Prozess des Designs der französischen Tastatur

Extended formulations for convex envelopes

In this work we derive explicit descriptions for the convex envelope of nonlinear functions that are component-wise concave on a subset of the variables and convex on the other variables. These functions account for more than 30% of all nonlinearities in common benchmark libraries. To overcome the combinatorial difficulties in deriving the convex envelope description given by the component-wise concave part of the functions, we consider an extended formulation of the convex envelope based on the Reformulation-Linearization-Technique introduced by Sherali and Adams(SIAM J Discret Math 3(3):411-430, 1990). Computational results are reported showing that the extended formulation strategy is a useful tool in global optimization

Semidefinite Programming. methods and algorithms for energy management

La présente thèse a pour objet d explorer les potentialités d une méthode prometteuse de l optimisation conique, la programmation semi-définie positive (SDP), pour les problèmes de management d énergie, à savoir relatifs à la satisfaction des équilibres offre-demande électrique et gazier.Nos travaux se déclinent selon deux axes. Tout d abord nous nous intéressons à l utilisation de la SDP pour produire des relaxations de problèmes combinatoires et quadratiques. Si une relaxation SDP dite standard peut être élaborée très simplement, il est généralement souhaitable de la renforcer par des coupes, pouvant être déterminées par l'étude de la structure du problème ou à l'aide de méthodes plus systématiques. Nous mettons en œuvre ces deux approches sur différentes modélisations du problème de planification des arrêts nucléaires, réputé pour sa difficulté combinatoire. Nous terminons sur ce sujet par une expérimentation de la hiérarchie de Lasserre, donnant lieu à une suite de SDP dont la valeur optimale tend vers la solution du problème initial.Le second axe de la thèse porte sur l'application de la SDP à la prise en compte de l'incertitude. Nous mettons en œuvre une approche originale dénommée optimisation distributionnellement robuste , pouvant être vue comme un compromis entre optimisation stochastique et optimisation robuste et menant à des approximations sous forme de SDP. Nous nous appliquons à estimer l'apport de cette approche sur un problème d'équilibre offre-demande avec incertitude. Puis, nous présentons une relaxation SDP pour les problèmes MISOCP. Cette relaxation se révèle être de très bonne qualité, tout en ne nécessitant qu un temps de calcul raisonnable. La SDP se confirme donc être une méthode d optimisation prometteuse qui offre de nombreuses opportunités d'innovation en management d énergie.The present thesis aims at exploring the potentialities of a powerful optimization technique, namely Semidefinite Programming, for addressing some difficult problems of energy management. We pursue two main objectives. The first one consists of using SDP to provide tight relaxations of combinatorial and quadratic problems. A first relaxation, called standard can be derived in a generic way but it is generally desirable to reinforce them, by means of tailor-made tools or in a systematic fashion. These two approaches are implemented on different models of the Nuclear Outages Scheduling Problem, a famous combinatorial problem. We conclude this topic by experimenting the Lasserre's hierarchy on this problem, leading to a sequence of semidefinite relaxations whose optimal values tends to the optimal value of the initial problem.The second objective deals with the use of SDP for the treatment of uncertainty. We investigate an original approach called distributionnally robust optimization , that can be seen as a compromise between stochastic and robust optimization and admits approximations under the form of a SDP. We compare the benefits of this method w.r.t classical approaches on a demand/supply equilibrium problem. Finally, we propose a scheme for deriving SDP relaxations of MISOCP and we report promising computational results indicating that the semidefinite relaxation improves significantly the continuous relaxation, while requiring a reasonable computational effort.SDP therefore proves to be a promising optimization method that offers great opportunities for innovation in energy management.PARIS11-SCD-Bib. électronique (914719901) / SudocSudocFranceF

SEMIDEFINITE PROGRAMMING BASED ALGORITHMS FOR THE SPARSEST CUT PROBLEM

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)In this paper we analyze a known relaxation for the Sparsest Cut problem based on positive semidefinite constraints, and we present a branch and bound algorithm and heuristics based on this relaxation. The relaxed formulation and the algorithms were tested on small and moderate sized instances. It leads to values very close to the optimum solution values. The exact algorithm could obtain solutions for small and moderate sized instances, and the best heuristics obtained optimum or near optimum solutions for all tested instances. The semidefinite relaxation gives a lower bound C/W and each heuristic produces a cut S with a ratio c(S)/omega(S) where either cs is at most a factor of C or omega(S) is at least a factor of W. We solved the semidefinite relaxation using a semi-infinite cut generation with a commercial linear programming package adapted to the sparsest cut problem. We showed that the proposed strategy leads to a better performance compared to the use of a. known semidefinite programming solver.45275100Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP