<p>This thesis consists of an introduction and sevenindependent, but closely related, papers which all deal withproblems in structural optimization. In particular, we considermodels and methods for global optimization of problems intopology design of discrete and continuum structures.</p><p>In the first four papers of the thesis the nonconvex problemof minimizing the weight of a truss structure subject to stressconstraints is considered. First itis shown that a certainsubclass of these problems can equivalently be cast as linearprograms and thus efficiently solved to global optimality.Thereafter, the behavior of a certain well-known perturbationtechnique is studied. It is concluded that, in practice, thistechnique can not guarantee that a global minimizer is found.Finally, a convergent continuous branch-and-bound method forglobal optimization of minimum weight problems with stress,displacement, and local buckling constraints is developed.Using this method, several problems taken from the literatureare solved with a proof of global optimality for the firsttime.</p><p>The last three papers of the thesis deal with topologyoptimization of discretized continuum structures. Theseproblems are usually modeled as mixed or pure nonlinear 0-1programs. First, the behavior of certain often usedpenalization methods for minimum compliance problems isstudied. It is concluded that these methods may fail to producea zero-one solution to the considered problem. To remedy this,a material interpolation scheme based on a rational functionsuch that compli- ance becomes a concave function is proposed.Finally, it is shown that a broad range of nonlinear 0-1topology optimization problems, including stress- anddisplacement-constrained minimum weight problems, canequivalently be modeled as linear mixed 0-1 programs. Thisresult implies that any of the standard methods available forgeneral linear integer programming can now be used on topologyoptimization problems.</p><p><b>Keywords:</b>topology optimization, global optimization,stress constraints, linear programming, mixed integerprogramming, branch-and-bound.</p>

<p>Using mathematical models with the purpose to understand and store knowlegde about a system is not a new field in science with early contributions dated back to, e.g., Kepler’s laws of planetary motion.</p><p>The aim is to obtain such a comprehensive predictive and quantitative knowledge about a phenomenon so that mathematical expressions or models can be used to forecast every relevant detail about that phenomenon. Such models can be used for reducing pollutions from car engines; prevent aviation incidents; or developing new therapeutic drugs. Models used to forecast, or predict, the behavior of a system are refered to predictive models. For such, the estimation problem aims to find one model and is well known and can be handeled by using standard methods for global nonlinear optimization.</p><p>Descriptive models are used to obtain and store quantitative knowledge of system. Estimation of descriptive models has not been much described by the literature so far; instead the methods used for predictive models have beed applied. Rather than finding one particular model, the parameter estimation for descriptive models aims to find every model that contains descriptive information about the system. Thus, the parameter estimation problem for descriptive models can not be stated as a standard optimization problem.</p><p>The main objective for this thesis is to propose methods for estimation of descriptive models. This is made by using methods for nonlinear optimization including both new and existing theory.</p>

Water supply system planning is concerned about the design of water supply infrastructure for distributing water from sources to users. Population growth, economic development and diminishing freshwater supplies are posing growing challenges for water supply system planning in many urban areas. Besides the need to exploit alternative water sources to the conventional surface and groundwater supplies, such as reclaimed water, a systematic point of view has to be taken for the efficient management of all potential water resources, so that issues of water supply, storage, treatment and reuse are not considered separately, but rather in the context of their interactions. The focus of this dissertation is to develop mathematical models and optimization algorithms for water supply system planning, where the interaction of different system components is explicitly considered. A deterministic nonlinear programming model is proposed at first to decide pipe and pump sizes in a regional water supply system for satisfying given potable and non-potable user demands over a certain planning horizon. A branch-and-bound algorithm based on the reformulation-linearization technique is then developed for solving the model to global optimality. To handle uncertainty in the planning process, a stochastic programming (SP) model and a robust optimization (RO) model are successively proposed to deal with random water supply and demand and the risk of facility failure, respectively. Both models attempt to make the decision of building some additional treatment and recharge facilities for recycling wastewater on-the-site. While the objective of the SP model is to minimize the total system design and expected operation cost, the RO model tries to achieve a favorable trade-off between system cost and system robustness, where the system robustness is defined in terms of meeting given user demands against the worst-case failure mode. The Benders decomposition method is then applied for solving both models by exploiting their special structure.

The subject of this work is image coding and restoration in the context of satellite imaging. Regardless of recent developments in image restoration techniques and embedded compression algorithms, the reconstructed image still suffers from coding artifacts making its quality evaluation difficult. The objective of the thesis is to improve the quality of the final image with the study of the optimal structure of decoding and restoration regarding to the properties of the acquisition and compression processes. More essentially, the aim of this work is to propose a reliable technique to address the optimal decoding-deconvolution-denoising problem in the objective of global optimization of the compression/restoration chain. The thesis is organized in three parts. The first part is a general introduction to the problematic addressed in this work. We then review a state-of-the-art of restoration and compression techniques for satellite imaging and we describe the current imaging chain used by the French Space Agency as this is the focus of the thesis. The second part is concerned with the global optimization of the satellite imaging chain. We propose an approach to estimate the theoretical distortion of the complete chain and we present, for three different configurations of coding/restoration, an algorithm to perform its minimization. Our second contribution is also focused on the study of the global chain but is more aimed to optimize the visual quality of the final image. We present numerical methods to improve the quality of the reconstructed image and we propose a novel imaging chain based on the image quality assessment results of these techniques. The last part of the thesis introduces a satellite imaging chain based on a new sampling approach. This approach is interesting in the context of satellite imaging as it allows transferring all the difficulties to the on-ground decoder. We recall the main theoretical results of this sampling technique and we present a satellite imaging chain based on this framework. We propose an algorithm to solve the reconstruction problem and we conclude by comparing the proposed chain to the one currently used by the CNES.

Globaliojo optimizavimo metodai, pagrįsti Lipšico rėžių apskaičiavimu, yra plačiai taikomi įvairių optimizavimo uždavinių sprendimui. Tačiau Lipšico metodai dažniausiai remiasi prielaida, kad Lipšico konstanta žinoma iš anksto, o tai retas atvejis sprendžiant praktinius uždavinius. Todėl Simonos Talačkaitės magistro darbe yra toliau nagrinėjama aktuali ir svarbi problematika iškylanti realizuojant Lipšico metodus nesiremiančius jokiomis išankstinėmis prielaidomis apie Lipšico konstantą. Praktinio tiriamojo pobūdžio magistro darbe iškeliamas toks pagrindinis tikslas: ištirti daugiamačių simpleksinių globaliojo optimizavimo algoritmų su nežinoma Lipšico konstanta efektyvumą priklausomai nuo naudojamo simplekso centro. Šiam tikslui pasiekti buvo iškelti šie uždaviniai: apžvelgti naujausią literatūrą skirta Lipšico metodams su nežinoma Lipšico konstanta; matematiškai išnagrinėti įvairių daugiamačių simplekso centrų apskaičiavimus bendru atveju bei juos realizuoti Matlab aplinkoje; papildyti simpleksinį globaliojo optimizavimo DISIMPL algoritmą šių simpleksų centrų apskaičiavimo paprogramėmis; eksperimentiškai ištirti pasiūlytų rezultatų praktiškumą sprendžiant testinius optimizavimo uždavinius. / This work analyzes Global optimization objectives, the most important it will be algorithms with simplicial Lipšico constant. Also, this work analyzes multidi- mensional DIRECT algorithm. We will provide dividing in higher dimennsions DIRECT algorithm. Then analyzes two simplex and apply the solutions. The hand simplex to smallerpartitions. Perceive multidimensional DIRECT algorithm division rules. In this work wrote a lot about simplicial center about dividing of hyoer-cube. Finally, the experiment it will be about the best way, how we can …nd circle center ir diferent way. Simplex centers using 8 test funkcions , changing the number of iterations and mistakes number. Create tables and to analyzes them. The purpose of this paper work is to introduce the simplex algorithm for global optimization with unknown Lipšicas constant depending on the e¢ ciency of the division of the rules used in the simplex.

An efficient robustness analysis for the VEGA launch vehicle is essential to minimize the potential system failure during the ascending phase. Monte Carlo sampling method is usually considered as a reliable strategy in industry if the sampling size is large enough. However, due to a large number of uncertainties and a long response time for a single simulation, exploring the entire uncertainties sufficiently through Monte Carlo sampling method is impractical for VEGA launch vehicle. In order to make the robustness analysis more efficient when the number of simulation is limited, the quasi-Monte Carlo(Sobol, Faure, Halton sequence) and heuristic algorithm(Differential Evolution) are proposed. Nevertheless, the reasonable number of samples for simulation is still much smaller than the minimal number of samples for sufficient exploration. To further improve the efficiency of robustness analysis, the redundant uncertainties are sorted out by sensitivity analysis. Only the dominant uncertainties are remained in the robustness analysis. As all samples for simulation are discrete, many uncertainty spaces are not explored with respect to its objective function by sampling or optimization methods. To study these latent information, the meta-model trained by Gaussian Process is introduced. Based on the meta-model, the expected maximum objective value and expected sensitivity of each uncertainties can be analyzed for robustness analysis with much higher efficiency but without loss much accuracy.

In telecommunications in general and in GSM in particular, the handover is a feature that guarantees a smooth transition of a call from one base station - that is for the purpose of this project an antenna - to another. In the recent ten years, the amount of data traffic through mobile telecommunications has doubled annually, putting an enormous strain on the network and forcing operators to upgrade with more and more base stations and new features. Although 3G and 4G are responsible for data traffic in most countries, GSM still provides more than 80% of the coverage for mobile devices around the world. Due to the increase in data traffic, 3G and 4G need to use more and more frequencies at the expense of GSM. An optimization of the GSM network is thus vital. In this project, we research two methods to automatically choose the parameters of interest (PoI) that govern the handover feature in each cell which is, roughly speaking, the area of coverage of one antenna. In one of these methods, the choice of cell- and cell-to-cell-specific parameters has its origins in control theory while the other method is based on mathematical optimization. In the mathematical sense, our goal is to optimize the quality of service over PoIs. Extensive simulations have been run using these PoIs in order to evaluate if and how the two different methods can effectively be used in reality. Several useful insights have been gained that will provide the basis for future work. The optimization approach in particular has proved to deliver good results within the limitations of the simulated environment used for testing.

L’«Efficient Global Optimization» (EGO) est une méthode de référence pour l’optimisation globale de fonctions «boites noires» coûteuses. Elle peut cependant rencontrer quelques difficultés, comme le mauvais conditionnement des matrices de covariance des processus Gaussiens (GP) qu’elle utilise, ou encore la lenteur de sa convergence vers l’optimum global. De plus, le choix des paramètres du GP, crucial car il contrôle la famille des fonctions d’approximation utilisées, mériterait une étude plus poussée que celle qui en a été faite jusqu’à présent. Enfin, on peut se demander si l’évaluation classique des paramètres du GP est la plus appropriée à des fins d’optimisation. \\Ce travail est consacré à l'analyse et au traitement des différentes questions soulevées ci-dessus.La première partie de cette thèse contribue à une meilleure compréhension théorique et pratique de l’impact des stratégies de régularisation des processus Gaussiens, développe une nouvelle technique de régularisation, et propose des règles pratiques. Une seconde partie présente un nouvel algorithme combinant EGO et CMA-ES (ce dernier étant un algorithme d’optimisation globale et convergeant). Le nouvel algorithme, nommé EGO-CMA, utilise EGO pour une exploration initiale, puis CMA-ES pour une convergence finale. EGO-CMA améliore les performances des deux algorithmes pris séparément. Dans une troisième partie, l’effet des paramètres du processus Gaussien sur les performances de EGO est soigneusement analysé. Finalement, un nouvel algorithme EGO auto-adaptatif est présenté, dans une nouvelle approche où ces paramètres sont estimés à partir de leur influence sur l’efficacité de l’optimisation elle-même. / The Efficient Global Optimization (EGO) is regarded as the state-of-the-art algorithm for global optimization of costly black-box functions. Nevertheless, the method has some difficulties such as the ill-conditioning of the GP covariance matrix and the slow convergence to the global optimum. The choice of the parameters of the GP is critical as it controls the functional family of surrogates used by EGO. The effect of different parameters on the performance of EGO needs further investigation. Finally, it is not clear that the way the GP is learned from data points in EGO is the most appropriate in the context of optimization. This work deals with the analysis and the treatment of these different issues. Firstly, this dissertation contributes to a better theoretical and practical understanding of the impact of regularization strategies on GPs and presents a new regularization approach based on distribution-wise GP. Moreover, practical guidelines for choosing a regularization strategy in GP regression are given. Secondly, a new optimization algorithm is introduced that combines EGO and CMA-ES which is a global but converging search. The new algorithm, called EGO-CMA, uses EGO for early exploration and then CMA-ES for final convergence. EGO-CMA improves the performance of both EGO and CMA-ES. Thirdly, the effect of GP parameters on the EGO performance is carefully analyzed. This analysis allows a deeper understanding of the influence of these parameters on the EGO iterates. Finally, a new self-adaptive EGO is presented. With the self-adaptive EGO, we introduce a novel approach for learning parameters directly from their contribution to the optimization.

Cette thèse intitulée « Une approche exacte de résolution de problèmes de pooling appliquée à la fabrication d’aliments », porte sur la résolution (par des méthodes exactes d’optimisation) de problèmes industriels liés à la fabrication d’aliments. Ces problèmes industriels traitent de l’aide à la décision pour la fabrication d’aliments pour des animaux et se rapprochent de problèmes biens connus de la littérature scientifique, à savoir les problèmes de pooling. La méthode présentée dans cet exposé permet de résoudre les problèmes d’optimisation bilinéaires issus de cette problématique industrielle. Elle est basée un branch-and-bound résolvant des linéarisations. Une approche lagrangienne a aussi été explorée et testée pour calculer des bornes inférieures. / « A global approach to solve pooling problem applied to feed mix industry » deals with the resolution of non linear non convex optimization problem which can occur in the feed mix industry. Feed mix industry problems are close to pooling problem, well-known in the literature. They are aimed to help decision maker in formulating feed, ie. To decide how to blend raw material to make a product satisfying nutrient and production constraints. The brand-and-bound algorithm presented in this these is aimed to solved large-scaled bilinear problems with bilinear constraints. A lagrangian approach has also been developed to obtain valid lower bound.

Aujourd'hui, l'alimentation représente plus de 70% du coût de production en engraissement porcin et dans le contexte économique actuel, il est important de parvenir à réduire ce coût. L'alimentation utilisée actuellement utilise des phases et est représentée par un modèle linéaire. L'alimentation par mélanges introduite récemment est représentée par un modèle bilinéaire. Nous introduisons ici une nouvelle formulation qui est une combinaison des alimentations traditionnelle par mélanges: la méthode hybride. Nous montrons qu'elle permet de réduire le coût de plus de 5%. L'étude principale porte sur l'optimisation globale du problème bilinéaire, non convexe, modélisant l'alimentation par mélanges. La bilinéarité apparaît dans la fonction objectif et dans les contraintes. Ce problème peut posséder plusieurs minima, mais nous souhaitons obtenir un minimum global. Il est équivalent à un problème de pooling et nous montrons qu'il est fortement NP-difficile. Après de premiers résultats, nous énonçons la conjecture que tout minimum local est global pour ce problème bilinéaire appliqué à l'industrie porcine. Nous la prouvons sur un exemple de dimension réduite. Notre problème ne pouvant pas être résolu avec des solveurs globaux à cause de sa dimension, nous utilisons des approches telle que la pénalisation, la discrétisation, et techniques de relaxation lagrangienne ou convexe. Toutes ces approches supportent notre conjecture. Nous faisons également une étude de la robustesse des modèles à la variation des prix des ingrédients ainsi qu'une étude multicritère nous permettant d'obtenir des résultats numériques réduisant considérablement les rejets, autres enjeux importants. / Today, feed represents more than 70% of the production cost in growing-finishing pig industry and in the current economic context, it is important to reduce it. The feeding system currently used uses phases and is expressed as a linear model. The feeding system using feeds introduced more recently is represented by a bilinear model. We introduced here new feeding system which is a combination offeeding systems using phases and feeds: the hybrid method. We show that it can reduce the feed cost by more than 5%. The main part of this manuscript is about global optimization of the bilinear problem, and non convex, problem modeling feeding system using feeds. The objective function and some constraints are bilinear. This problem can have several local minima but we would like to have a global one. It is equivalent to a pooling problem and we prove that it is a strongly NP-hard problem. After a study of first results, we enounce the conjecture that any local minimum is a global minimum for that problem applied in the pig industry. We prove it for a small size example. Our problem cannot be solved by using global solver due to its size, then we applied some relaxation methods such as penalization of bilinear terms, their discretization and Langrangian and convex relaxations. All these approaches support our conjecture. Then we study the robustness of the models on the ingredient price variations and a multicriteria study reducing phosphorus and nitrogen excretion.