A compact variant of the QCR method for quadratically constrained quadratic 0-1 programs

Quadratic Convex Reformulation (QCR) is a technique that was originally proposed for quadratic 0-1 programs, and then extended to various other problems. It is used to convert non-convex instances into convex ones, in such a way that the bound obtained by solving the continuous relaxation of the reformulated instance is as strong as possible. In this paper, we focus on the case of quadratically constrained quadratic 0-1 programs. The variant of QCR previously proposed for this case involves the addition of a quadratic number of auxiliary continuous variables. We show that, in fact, at most one additional variable is needed. Some computational results are also presented

Sustainable two stage supply chain management: A quadratic optimization approach with a quadratic constraint

Designing a supply chain to comply with environmental policy requires awareness of how work and/or production methods impact the environment and what needs to be done to reduce those environmental impacts and make the company more sustainable. This is a dynamic process that occurs at both the strategic and operational levels. However, being environmentally friendly does not necessarily mean improving the efficiency of the system at the same time. Therefore, when allocating a production budget in a supply chain that implements the green paradigm, it is necessary to figure out how to properly recover costs in order to improve both sustainability and routine operations, offsetting the negative environmental impact of logistics and production without compromising the efficiency of the processes to be executed. In this paper, we study the latter problem in detail, focusing on the CO2 emissions generated by the transportation from suppliers to production sites, and by the production activities carried out in each plant. We do this using a novel mathematical model that has a quadratic objective function and all linear constraints except one, which is also quadratic, and models the constraint on the budget that can be used for green investments caused by the increasing internal complexity created by large production flows in the production nodes of the supply network. To solve this model, we propose a multistart algorithm based on successive linear approximations. Computational results show the effectiveness of our proposal

Regret Models and Preprocessing Techniques for Combinatorial Optimization under Uncertainty

Ph.DDOCTOR OF PHILOSOPH

Enhancements of Discretization Approaches for Non-Convex Mixed-Integer Quadratically Constraint Quadratic Programming

We study mixed-integer programming (MIP) relaxation techniques for the solution of non-convex mixed-integer quadratically constrained quadratic programs (MIQCQPs). We present two MIP relaxation methods for non-convex continuous variable products that enhance existing approaches. One is based on a separable reformulation, while the other extends the well-known MIP relaxation normalized multiparametric disaggregation technique (NMDT). In addition, we introduce a logarithmic MIP relaxation for univariate quadratic terms, called sawtooth relaxation, based on [4]. We combine the latter with the separable reformulation to derive MIP relaxations of MIQCQPs. We provide a comprehensive theoretical analysis of these techniques, and perform a broad computational study to demonstrate the effectiveness of the enhanced MIP relaxations in terms producing tight dual bounds for MIQCQP

On Minimal Valid Inequalities for Mixed Integer Conic Programs

We study disjunctive conic sets involving a general regular (closed, convex, full dimensional, and pointed) cone K such as the nonnegative orthant, the Lorentz cone or the positive semidefinite cone. In a unified framework, we introduce K-minimal inequalities and show that under mild assumptions, these inequalities together with the trivial cone-implied inequalities are sufficient to describe the convex hull. We study the properties of K-minimal inequalities by establishing algebraic necessary conditions for an inequality to be K-minimal. This characterization leads to a broader algebraically defined class of K- sublinear inequalities. We establish a close connection between K-sublinear inequalities and the support functions of sets with a particular structure. This connection results in practical ways of showing that a given inequality is K-sublinear and K-minimal. Our framework generalizes some of the results from the mixed integer linear case. It is well known that the minimal inequalities for mixed integer linear programs are generated by sublinear (positively homogeneous, subadditive and convex) functions that are also piecewise linear. This result is easily recovered by our analysis. Whenever possible we highlight the connections to the existing literature. However, our study unveils that such a cut generating function view treating the data associated with each individual variable independently is not possible in the case of general cones other than nonnegative orthant, even when the cone involved is the Lorentz cone

Mixed-integer Nonlinear Optimization: a hatchery for modern mathematics

The second MFO Oberwolfach Workshop on Mixed-Integer Nonlinear Programming (MINLP) took place between 2nd and 8th June 2019. MINLP refers to one of the hardest Mathematical Programming (MP) problem classes, involving both nonlinear functions as well as continuous and integer decision variables. MP is a formal language for describing optimization problems, and is traditionally part of Operations Research (OR), which is itself at the intersection of mathematics, computer science, engineering and econometrics. The scientific program has covered the three announced areas (hierarchies of approximation, mixed-integer nonlinear optimal control, and dealing with uncertainties) with a variety of tutorials, talks, short research announcements, and a special "open problems'' session

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

Nondeterministic hybrid dynamical systems

This thesis is concerned with the analysis, control and identification of hybrid dynamical systems. The main focus is on a particular class of hybrid systems consisting of linear subsystems. The discrete dynamic, i.e., the change between subsystems, is unknown or nondeterministic and cannot be influenced, i.e. controlled, directly. However changes in the discrete dynamic can be detected immediately, such that the current dynamic (subsystem) is known. In order to motivate the study of hybrid systems and show the merits of hybrid control theory, an example is given. It is shown that real world systems like Anti Locking Brakes (ABS) are naturally modelled by such a class of linear hybrids systems. It is shown that purely continuous feedback is not suitable since it cannot achieve maximum braking performance. A hybrid control strategy, which overcomes this problem, is presented. For this class of linear hybrid system with unknown discrete dynamic, a framework for robust control is established. The analysis methodology developed gives a robustness radius such that the stability under parameter variations can be analysed. The controller synthesis procedure is illustrated in a practical example where the control for an active suspension of a car is designed. Optimal control for this class of hybrid system is introduced. It is shows how a control law is obtained which minimises a quadratic performance index. The synthesis procedure is stated in terms of a convex optimisation problem using linear matrix inequalities (LMI). The solution of the LMI not only returns the controller but also the performance bound. Since the proposed controller structures require knowledge of the continuous state, an observer design is proposed. It is shown that the estimation error converges quadratically while minimising the covariance of the estimation error. This is similar to the Kalman filter for discrete or continuous time systems. Further, we show that the synthesis of the observer can be cast into an LMI, which conveniently solves the synthesis problem