Sum-Rate Maximization in Two-Way AF MIMO Relaying: Polynomial Time Solutions to a Class of DC Programming Problems

Sum-rate maximization in two-way amplify-and-forward (AF) multiple-input multiple-output (MIMO) relaying belongs to the class of difference-of-convex functions (DC) programming problems. DC programming problems occur as well in other signal processing applications and are typically solved using different modifications of the branch-and-bound method. This method, however, does not have any polynomial time complexity guarantees. In this paper, we show that a class of DC programming problems, to which the sum-rate maximization in two-way MIMO relaying belongs, can be solved very efficiently in polynomial time, and develop two algorithms. The objective function of the problem is represented as a product of quadratic ratios and parameterized so that its convex part (versus the concave part) contains only one (or two) optimization variables. One of the algorithms is called POlynomial-Time DC (POTDC) and is based on semi-definite programming (SDP) relaxation, linearization, and an iterative search over a single parameter. The other algorithm is called RAte-maximization via Generalized EigenvectorS (RAGES) and is based on the generalized eigenvectors method and an iterative search over two (or one, in its approximate version) optimization variables. We also derive an upper-bound for the optimal values of the corresponding optimization problem and show by simulations that this upper-bound can be achieved by both algorithms. The proposed methods for maximizing the sum-rate in the two-way AF MIMO relaying system are shown to be superior to other state-of-the-art algorithms.Comment: 35 pages, 10 figures, Submitted to the IEEE Trans. Signal Processing in Nov. 201

Adaptive exact penalty DC algorithms for nonsmooth DC optimization problems with equality and inequality constraints

We propose and study two DC (difference of convex functions) algorithms based on exact penalty functions for solving nonsmooth DC optimization problems with nonsmooth DC equality and inequality constraints. Both methods employ adaptive penalty updating strategies to improve their performance. The first method is based on exact penalty functions with individual penalty parameter for each constraint (i.e. multidimensional penalty parameter) and utilizes a primal-dual approach to penalty updates. The second method is based on the so-called steering exact penalty methodology and relies on solving some auxiliary convex subproblems to determine a suitable value of the penalty parameter. We present a detailed convergence analysis of both methods and give several simple numerical examples highlighting peculiarites of two different penalty updating strategies studied in this paper

Global optimization at work

In many research situations where mathematical models are used, researchers try to find parameter values such that a given performance criterion is at an optimum. If the parameters can be varied in a continuous way, this in general defines a so-called Nonlinear Programming Problem. Methods for Nonlinear Programming usually result in local optima. A local optimum is a solution (parameter values) which is the best with respect to values in the neighbourhood of that solution, not necessarily the best over the total admissible, feasible set of all possible parameter values, solutions.For mathematicians this results in the research question: How to find the best, global optimum in situations where several local optima exist?, the field of Global Optimization (GLOP). Literature, books and a specific journal, has appeared during the last decades on the field. Main focus has been on the mathematical side, i.e. given assumptions on the structure of the problems to be solved and specific global optimization methods and properties are derived. Cooperation between mathematicians and researchers (in this book called 'the modeller' or 'the potential user'), who saw global optimization problems in practical problems has lead to application of GLOP algorithms to practical optimization problems. Some of those can be found in this book. In this book we started with the question:Given a potential user with an arbitrary global optimization problem, what route can be taken in the GLOP forest to find solutions of the problem?From this first question we proceed by raising new questions. In Chapter 1 we outline the target group of users we have in mind, i.e. agricultural and environmental engineers, designers and OR workers in agricultural science. These groups are not clearly defined, nor mutually exclusive, but have in common that mathematical modelling is used and there is knowledge of linear programming and possibly of combinatorial optimization.In general, when modellers are confronted with optimization aspects, the first approach is to develop heuristics or to look for standard nonlinear programming codes to generate solutions of the optimization problem. During the search for solutions, multiple local optima may appear. We distinguished two major tracks for the path to be taken from there by the potential user to solve the problem. One track is called the deterministic track and is discussed in Chapters 2, 3 and 4. The other track is called the stochastic track and is discussed in Chapters 5 and 6. The two approaches are intended to reach a different goal.The deterministic track aims at:The global optimum is approximated (found) with certainty in a finite number of steps.The stochastic track is understood to contain some stochastic elements and aims at:Approaching the optimum in a probabilistic sense as effort grows to infinity.Both tracks are investigated in this book from the viewpoint of a potential user corresponding to the way of thinking in Popperian science. The final results are new challenging problems, questions for further research. A side question along the way is:How can the user influence the search process given the knowledge of the underlying problem and the information that becomes available during the search?The deterministic approachWhen one starts looking into the deterministic track for a given problem, one runs into the requirements which determine a major difference in applicability of the two approaches.Deterministic methods require the availability of explicit mathematical expressions of the functions to be optimized.In many practical situations which are also discussed in this book, these expressions are not available and deterministic methods cannot be applied. The operations in deterministic methods are based on concepts such as Branch-and-Bound and Cutting which require bounding of functions and parameters based on so-called mathematical structures.In Chapter 2 we describe these structures and distinguish between those which can be derived directly from the expressions, such as quadratic, bilinear and fractional functions and other structures which require analysis of the expressions such as concave and Lipschitz continuous functions. Examples are given of optimization problems revealing their structure. Moreover, we show that symmetry in the model formulation may cause models to have more than one extreme.In Chapter 3 the relationship between GLOP and Integer Programming (IP) is highlighted for several reasons.Sometimes practical GLOP problems can be approximated by IP variants and solved by standard Mixed Integer Linear Programming (MILP) techniques.The algorithms of GLOP and IP can similarly be classified.The transformability of GLOP problems to IP problems and vice versa shows that difficult problems in one class will not become easier to solve in the other.Analysis of problems, which is common in Global Optimization, can be used to better understand the complexity of some IP problems.In Chapter 4 we analyze the use of deterministic methods, demonstrating the application of the Branch-and-Bound concept. The following can be stated from the point of view of the potential user:Analysis of the expressions is required to find useful mathematical structures (Chapter 2). It should be noted that also interval arithmetic techniques can be applied directly on the expressions.The elegance of the techniques is the guarantee that we are certain about the global optimality of the optimum, when it has been discovered and verified.The methods are hard to implement. Thorough use should be made of special data structures to store the necessary information in memory.Two cases are elaborated. The quadratic product design problem illustrates how the level of Decision Support Systems can be reached for low dimensional problems, i.e. the number of variables, components or ingredients, is less than 10. The other case, the nutrient problem, shows how by analysis of the problem many useful properties can be derived which help to cut away large areas of the feasible space where the optimum cannot be situated. However, it also demonstrates the so-called Curse of Dimensionality; the problem has so many variables in a realistic situation that it is impossible to traverse the complete Branch-and-Bound tree. Therefore it is good to see the relativity of the use of deterministic methods:No global optimization method can guarantee to find and verify the global optimum for every practical situation, within a humans lifetime.The stochastic approachThe stochastic approach is followed in practice for many optimization problems by combining the generation of random points with standard nonlinear optimization algorithms. The following can be said from the point of view of the potential user.The methods require no mathematical structure of the problem and are therefore more generally applicable.The methods are relatively easy to implement.The user is never completely certain that the global optimum has been reached.The optimum is approximated in a probabilistic sense when effort increases to infinity.In Chapter 5 much attention is paid to the question what happens when a user wants to spend a limited (not infinite) amount of time to the search for the optimum, preferably less than a humans lifetime:What to do when the time for solving the problem is finite?First we looked at the information which becomes available during the search and the instruments with which the user can influence the search. It appeared that besides classical instruments which are also available in traditional nonlinear programming, the main instrument is to influence the trade-off between global (random) and local search (looking for a local optimum). This lead to a new question:Is there a best way to rule the choice between global and local search, given the information which becomes available?Analyzing in a mathematical way with extreme cases lead to the comfortable conclusion that a best method of choosing between global and local search -thus a best global optimization method- does not exist. This is valid for cases where further information (more than the information which becomes available during the search) on the function to be optimized is not available, called in literature the black-box case. The conclusion again shows that mathematical analysis with extreme cases is a powerful tool to demonstrate that so-called magic algorithms -algorithms which are said in scientific journals to be very promising, because they perform well on some test cases- can be analyzed and 'falsified' in the way of Popperian thinking. This leads to the conclusion that:Magic algorithms which are going to solve all of your problems do not exist.Several side questions derived from the main problem are investigated in this book.In Chapter 6 we place the optimization problem in the context of parameter estimation. One practical question is raised by the phenomenonEvery local search leads to a new local optimum.We know from parameter estimation that this is a symptom in so called non-identifiable systems. The minimum is obtained at a lower dimensional surface or curve. Some (non-magic) heuristics are discussed to overcome this problem.There are two side questions of users derived from the general remark:"I am not interested in the best (GLOP) solution, but in good points".The first question is that of Robust Solutions, introduced in Chapter 4, and the other is called Uniform Covering, concerning the generation of points which are nearly as good as the optimum, discussed in Chapter 6.Robust solutions are discussed in the context of product design. The robustness is defined as a measure of the error one can make from the solution so that the solution (product) is still acceptable. Looking for the most robust product is looking for that point which is as far away as possible from the boundaries of the feasible (acceptable) area. For the solution procedures, we had a look at the appearance of the problem in practice, where boundaries are given by linear and quadratic surfaces, properties of the product.For linear boundaries, finding the most robust solution is an LP problem and thus rather easy.For quadratic properties the development of specific algorithms is required.The question of Uniform Covering concerns the desire to have a set of "suboptimal" points, i.e. points with low function value (given an upper level of the function value); the points are in a so-called level set. To generate "low" points, one could run a local search many times. However, we want the points not to be concentrated in one of the compartments or one sub-area of the level set, we want them to be equally, uniformly spread over the region. This is a very difficult problem for which we test and analyze several approaches in Chapter 6. The analysis taught us that:It is unlikely that stochastic methods will be proposed which solve problems in an expected calculation time, which is polynomial in the number of variables of the problem.Final resultWhether an arbitrary problem of a user can be solved by GLOP requires analysis. There are many optimization problems which can be solved satisfactorily. Besides the selection of algorithms the user has various instruments to steer the process. For stochastic methods it mainly concerns the trade-off between local and global search. For deterministic methods it includes setting bounds and influencing the selection rule in Branch-and-Bound. We hope with this book to have given a tool and a guidance to solution procedures. Moreover, it is an introduction to further literature on the subject of Global Optimization.</p

Global optimisation of large-scale quadratic programs: application to short-term planning of industrial refinery-petrochemical complexes

This thesis is driven by an industrial problem arising in the short-term planning of an integrated refinery-petrochemical complex (IRPC) in Colombia. The IRPC of interest is composed of 60 industrial plants and a tank farm for crude mixing and fuel blending consisting of 30 additional units. It considers both domestic and imported crude oil supply, as well as refined product imports such as low sulphur diesel and alkylate. This gives rise to a large-scale mixed-integer quadratically constrained quadratic program (MIQCQP) comprising about 7,000 equality constraints with over 35,000 bilinear terms and 280 binary variables describing operating modes for the process units. Four realistic planning scenarios are recreated to study the performance of the algorithms developed through the thesis and compare them to commercial solvers. Local solvers such as SBB and DICOPT cannot reliably solve such large-scale MIQCQPs. Usually, it is challenging to even reach a feasible solution with these solvers, and a heuristic procedure is required to initialize the search. On the other hand, global solvers such as ANTIGONE and BARON determine a feasible solution for all the scenarios analysed, but they are unable to close the relaxation gap to less than 40% on average after 10h of CPU runtime. Overall, this industrial-size problem is thus intractable to global optimality in a monolithic way. The first main contribution of the thesis is a deterministic global optimisation algorithm based on cluster decomposition (CL) that divides the network into groups of process units according to their functionality. The algorithm runs through the sequences of clusters and proceeds by alternating between: (i) the (global) solution of a mixed-integer linear program (MILP), obtained by relaxing the bilinear terms based on their piecewise McCormick envelopes and a dynamic partition of their variable ranges, in order to determine an upper bound on the maximal profit; and (ii) the local solution of a quadratically-constrained quadratic program (QCQP), after fixing the binary variables and initializing the continuous variables to the relaxed MILP solution point, in order to determine a feasible solution (lower bound on the maximal profit). Applied to the base case scenario, the CL approach reaches a best solution of 2.964 MMUSD/day and a relaxation gap of 7.5%, a remarkable result for such challenging MIQCQP problem. The CL approach also vastly outperforms both ANTIGONE (2.634 MMUSD/day, 32% optimality gap) and BARON (2.687 MMUSD/day, 40% optimality gap). The second main contribution is a spatial Lagrangean decomposition, which entails decomposing the IRPC short-term planning problem into a collection of smaller subproblems that can be solved independently to determine an upper bound on the maximal profit. One advantage of this strategy is that each sub-problem can be solved to global optimality, potentially providing good initial points for the monolithic problem itself. It furthermore creates a virtual market for trading crude blends and intermediate refined–petrochemical streams and seeks an optimal trade-off in such a market, with the Lagrange multipliers acting as transfer prices. A decomposition over two to four is considered, which matches the crude management, refinery, petrochemical operations, and fuel blending sections of the IRPC. An optimality gap below 4% is achieved in all four scenarios considered, which is a significant improvement over the cluster decomposition algorithm.Open Acces

Programmation DC et DCA pour l'optimisation non convexe/optimisation globale en variables mixtes entières (Codes et Applications)

Basés sur les outils théoriques et algorithmiques de la programmation DC et DCA, les travaux de recherche dans cette thèse portent sur les approches locales et globales pour l'optimisation non convexe et l'optimisation globale en variables mixtes entières. La thèse comporte 5 chapitres. Le premier chapitre présente les fondements de la programmation DC et DCA, et techniques de Séparation et Evaluation (B&B) (utilisant la technique de relaxation DC pour le calcul des bornes inférieures de la valeur optimale) pour l'optimisation globale. Y figure aussi des résultats concernant la pénalisation exacte pour la programmation en variables mixtes entières. Le deuxième chapitre est consacré au développement d'une méthode DCA pour la résolution d'une classe NP-difficile des programmes non convexes non linéaires en variables mixtes entières. Ces problèmes d'optimisation non convexe sont tout d'abord reformulées comme des programmes DC via les techniques de pénalisation en programmation DC de manière que les programmes DC résultants soient efficacement résolus par DCA et B&B bien adaptés. Comme première application en optimisation financière, nous avons modélisé le problème de gestion de portefeuille sous le coût de transaction concave et appliqué DCA et B&B à sa résolution. Dans le chapitre suivant nous étudions la modélisation du problème de minimisation du coût de transaction non convexe discontinu en gestion de portefeuille sous deux formes : la première est un programme DC obtenu en approximant la fonction objectif du problème original par une fonction DC polyèdrale et la deuxième est un programme DC mixte 0-1 équivalent. Et nous présentons DCA, B&B, et l'algorithme combiné DCA-B&B pour leur résolution. Le chapitre 4 étudie la résolution exacte du problème multi-objectif en variables mixtes binaires et présente deux applications concrètes de la méthode proposée. Nous nous intéressons dans le dernier chapitre à ces deux problématiques challenging : le problème de moindres carrés linéaires en variables entières bornées et celui de factorisation en matrices non négatives (Nonnegative Matrix Factorization (NMF)). La méthode NMF est particulièrement importante de par ses nombreuses et diverses applications tandis que les applications importantes du premier se trouvent en télécommunication. Les simulations numériques montrent la robustesse, rapidité (donc scalabilité), performance et la globalité de DCA par rapport aux méthodes existantes.Based on theoretical and algorithmic tools of DC programming and DCA, the research in this thesis focus on the local and global approaches for non convex optimization and global mixed integer optimization. The thesis consists of 5 chapters. The first chapter presents fundamentals of DC programming and DCA, and techniques of Branch and Bound method (B&B) for global optimization (using the DC relaxation technique for calculating lower bounds of the optimal value). It shall include results concerning the exact penalty technique in mixed integer programming. The second chapter is devoted of a DCA method for solving a class of NP-hard nonconvex nonlinear mixed integer programs. These nonconvex problems are firstly reformulated as DC programs via penalty techniques in DC programming so that the resulting DC programs are effectively solved by DCA and B&B well adapted. As a first application in financial optimization, we modeled the problem pf portfolio selection under concave transaction costs and applied DCA and B&B to its solutions. In the next chapter we study the modeling of the problem of minimization of nonconvex discontinuous transaction costs in portfolio selection in two forms: the first is a DC program obtained by approximating the objective function of the original problem by a DC polyhedral function and the second is an equivalent mixed 0-1 DC program. And we present DCA, B&B algorithm, and a combined DCA-B&B algorithm for their solutions. Chapter 4 studied the exact solution for the multi-objective mixed zero-one linear programming problem and presents two practical applications of proposed method. We are interested int the last chapter two challenging problems: the linear integer least squares problem and the Nonnegative Mattrix Factorization problem (NMF). The NMF method is particularly important because of its many various applications of the first are in telecommunications. The numerical simulations show the robustness, speed (thus scalability), performance, and the globality of DCA in comparison to existent methods.ROUEN-INSA Madrillet (765752301) / SudocSudocFranceF

Applications of non-convex optimization in portfolio selection

Die vorgelegte Arbeit befasst sich mit nicht-konvexer Optimierung in dem Gebiet der Portfolio Selection. Thematisch lässt sich die Arbeit in zwei Teilgebiete strukturieren: (1) Das Lösen von Mean-Risk Problemen mit Value-at-Risk als Risikomaß: Es werden Methoden zum Auffinden von effizienten Portfolios für den Fall von diskret verteilten Asset Returns vorgestellt. Die behandelten Probleme sind (wegen der Nicht-Konvexität des Value-at-Risk) nicht konvex und lassen sich als Differenz von konvexen Funktionen darstellen. Es werden sowohl Branch-and-Bound als auch approximative Lösungsverfahren angewandt. Die globalen Lösungen des Branch-and-Bound werden mit den Lösungen der approximativen Verfahren verglichen. (2) Robustifizierung von Portfolio-Selection Problemen: In den letzten Jahren gibt es in der Literatur verstärkt Bemühungen Optimierungsprobleme bezüglich Unsicherheiten in den Parametern zu robustifizieren. Robustifizierte Lösungen haben die Eigenschaft, dass moderate Variationen von Parametern nicht zu dramatischen Verschlechterungen der Lösungen führen. Im Rahmen der robusten Portfolio Optimierung geht es hauptsächlich darum, Lösungen in Bezug auf Abweichungen in den Verteilungen der Gewinne der verwendeten Finanzinstrumente zu kontrollieren. In der gegenständlichen Arbeit werden mit Hilfe von Wahrscheinlichkeitsmetriken sogenannte Ambiguity Mengen definiert, welche alle Verteilungen enthalten, die aufgrund der Datenlage als mögliche Verteilungen in Frage kommen. Die verwendete Metrik, die sogenannte Kantorovich (Wasserstein) Metrik, ermöglicht es mittels Ergebnissen der nichtparametrischen Statistik, die Ambiguity Mengen als Konfidenzmengen um die empirischen Verteilungschätzer zu interpretieren. Mittels der beschriebenen Methoden werden Mean-Risk Probleme robustifiziert. Diese Probleme sind zunächst infinit und werden in einem weiteren Schritt zu nicht konvexen semi-definiten Problemen umformuliert. Die Lösung dieser Probleme basiert einerseits auf einem Algortihmus zum Lösen von semi-definiten Problemen mit unendlich vielen Nebenbedingungen und andererseits auf Methoden zum approximativen Lösen von nicht konvexen Problemen (dem sogenannten Difference of Convex Algorithm).The thesis is concerned with application of non-convex programming to problems of portfolio optimization in a single stage stochastic optimization framework. In particular two different classes of portfolio selection problems are investigated. In both the problems a scenario based approach to modeling uncertainty is pursued, i.e. the randomness in the models is always described by finitely many joint realizations of the asset returns. The thesis is structured into three chapters briefly outlined below: (1) A D.C. Formulation of Value-at-Risk constrained Optimization: In this Chapter the aim is to solve mean risk models with the Value-at-Risk as a risk measure. In the case of finitely supported return distributions, it is shown that the Value-at-Risk can be written as a D.C. function and the mentioned mean risk problem therefore corresponds to a D.C. problem. The non-convex problem of optimizing the Value at Risk is rather extensively treated in the literature and there are various approximative solution techniques as well as some approaches to solve the problem globally. The reformulation as D.C. problem provides an insight into the structure of the problem, which can be exploited to devise a Branch-and-Bound algorithm for finding global solutions for small to medium sized instances. The possibility of refining epsilon-optimal solutions obtained from the Branch-and-Bound framework via local search heuristics is also discussed in this Chapter. (2) Value-at-Risk constrained optimization using the DCA: In this part of the thesis the Value-at-Risk problem is once again investigated with the aim of solving problems of realistic sizes in relatively short time. Since the Value at Risk optimization can be shown to be a NP hard problem, this can only be achieved by sacrificing on the guaranteed globality of the solutions. Therefore a local solution technique for unconstrained D.C. problems called Difference of Convex Algorithm (DCA) is employed. To solve the problem a new variant of the DCA the so called 'hybrid DCA' is proposed, which preserves the favorable convergence properties of the computationally hard 'complete DCA' as well as the computational tractability of the so called 'simple DCA'. The results are tested for small problems and the solutions are shown to actually coincide with the global optima obtained with the Branch-and-Bound algorithm in most of the cases. For realistic problem sizes the proposed method is shown to consistently outperform known heuristic approximations implemented in commercial software. (3) A Framework for Optimization under Ambiguity: The last part of the thesis is devoted to a different topic which received much attention in the recent stochastic programming literature: the topic of robust optimization. More specifically the aim is to robustify single stage stochastic optimization models with respect to uncertainty about the distributions of the random variables involved in the formulation of the stochastic program. The aim is to explore ways of explicitly taking into account ambiguity about the distributions when finding a decision while imposing only very weak restrictions on possible probability models that are taken into consideration. Ambiguity is defined as possible deviation from a discrete reference measure Q (in this work the empirical measure). To this end a so called ambiguity set B, that contains all the measures that can reasonably be assumed to be the real measure P given the available data, is defined. Since the idea is to devise a general approach not restricted by assuming P to be an element of any specific parametric family, we define our ambiguity sets by the use of general probability metrics. Relative to these measures a worst case approach is adopted to robustify the problem with respect to B. The resulting optimization problems turn out to be infinite and are reduced to non-convex semi-definite problems. In the last part of the paper we show how to solve these problems numerically for the example of a mean risk portfolio selection problem with Expected Shortfall under a Threshold as the risk measure. The DCA in combination with an iterative algorithm to approximate the infinite set of constraints by finitely many ones is used to obtain numerical solutions to the problem

Proceedings of the XIII Global Optimization Workshop: GOW'16

[Excerpt] Preface: Past Global Optimization Workshop shave been held in Sopron (1985 and 1990), Szeged (WGO, 1995), Florence (GO’99, 1999), Hanmer Springs (Let’s GO, 2001), Santorini (Frontiers in GO, 2003), San José (Go’05, 2005), Mykonos (AGO’07, 2007), Skukuza (SAGO’08, 2008), Toulouse (TOGO’10, 2010), Natal (NAGO’12, 2012) and Málaga (MAGO’14, 2014) with the aim of stimulating discussion between senior and junior researchers on the topic of Global Optimization. In 2016, the XIII Global Optimization Workshop (GOW’16) takes place in Braga and is organized by three researchers from the University of Minho. Two of them belong to the Systems Engineering and Operational Research Group from the Algoritmi Research Centre and the other to the Statistics, Applied Probability and Operational Research Group from the Centre of Mathematics. The event received more than 50 submissions from 15 countries from Europe, South America and North America. We want to express our gratitude to the invited speaker Panos Pardalos for accepting the invitation and sharing his expertise, helping us to meet the workshop objectives. GOW’16 would not have been possible without the valuable contribution from the authors and the International Scientific Committee members. We thank you all. This proceedings book intends to present an overview of the topics that will be addressed in the workshop with the goal of contributing to interesting and fruitful discussions between the authors and participants. After the event, high quality papers can be submitted to a special issue of the Journal of Global Optimization dedicated to the workshop. [...

Nonconvex and mixed integer multiobjective optimization with an application to decision uncertainty

Multiobjective optimization problems commonly arise in different fields like economics or engineering. In general, when dealing with several conflicting objective functions, there is an infinite number of optimal solutions which cannot usually be determined analytically. This thesis presents new branch-and-bound-based approaches for computing the globally optimal solutions of multiobjective optimization problems of various types. New algorithms are proposed for smooth multiobjective nonconvex optimization problems with convex constraints as well as for multiobjective mixed integer convex optimization problems. Both algorithms guarantee a certain accuracy of the computed solutions, and belong to the first deterministic algorithms within their class of optimization problems. Additionally, a new approach to compute a covering of the optimal solution set of multiobjective optimization problems with decision uncertainty is presented. The three new algorithms are tested numerically. The results are evaluated in this thesis as well. The branch-and-bound based algorithms deal with box partitions and use selection rules, discarding tests and termination criteria. The discarding tests are the most important aspect, as they give criteria whether a box can be discarded as it does not contain any optimal solution. We present discarding tests which combine techniques from global single objective optimization with outer approximation techniques from multiobjective convex optimization and with the concept of local upper bounds from multiobjective combinatorial optimization. The new discarding tests aim to find appropriate lower bounds of subsets of the image set in order to compare them with known upper bounds numerically.Multikriterielle Optimierungprobleme sind in diversen Anwendungsgebieten wie beispielsweise in den Wirtschafts- oder Ingenieurwissenschaften zu finden. Da hierbei mehrere konkurrierende Zielfunktionen auftreten, ist die Lösungsmenge eines derartigen Optimierungsproblems im Allgemeinen unendlich groß und kann meist nicht in analytischer Form berechnet werden. In dieser Dissertation werden neue Branch-and-Bound basierte Algorithmen zur Lösung verschiedener Klassen von multikriteriellen Optimierungsproblemen entwickelt und vorgestellt. Der Branch-and-Bound Ansatz ist eine typische Methode der globalen Optimierung. Einer der neuen Algorithmen löst glatte multikriterielle nichtkonvexe Optimierungsprobleme mit konvexen Nebenbedingungen, während ein zweiter zur Lösung multikriterieller gemischt-ganzzahliger konvexer Optimierungsprobleme dient. Beide Algorithmen garantieren eine gewisse Genauigkeit der berechneten Lösungen und gehören damit zu den ersten deterministischen Algorithmen ihrer Art. Zusätzlich wird ein Algorithmus zur Berechnung einer Überdeckung der Lösungsmenge multikriterieller Optimierungsprobleme mit Entscheidungsunsicherheit vorgestellt. Alle drei Algorithmen wurden numerisch getestet. Die Ergebnisse werden ebenfalls in dieser Arbeit ausgewertet. Die neuen Algorithmen arbeiten alle mit Boxunterteilungen und nutzen Auswahlregeln, sowie Verwerfungs- und Terminierungskriterien. Dabei spielen gute Verwerfungskriterien eine zentrale Rolle. Diese entscheiden, ob eine Box verworfen werden kann, da diese sicher keine Optimallösung enthält. Die neuen Verwerfungskriterien nutzen Methoden aus der globalen skalarwertigen Optimierung, Approximationstechniken aus der multikriteriellen konvexen Optimierung sowie ein Konzept aus der kombinatorischen Optimierung. Dabei werden stets untere Schranken der Bildmengen konstruiert, die mit bisher berechneten oberen Schranken numerisch verglichen werden können

Derivative free algorithms for nonsmooth and global optimization with application in cluster analysis

This thesis is devoted to the development of algorithms for solving nonsmooth nonconvex problems. Some of these algorithms are derivative free methods.Doctor of Philosoph

Optimization with Constraint Learning: A Framework and Survey

Many real-life optimization problems frequently contain one or more constraints or objectives for which there are no explicit formulas. If data is however available, these data can be used to learn the constraints. The benefits of this approach are clearly seen, however there is a need for this process to be carried out in a structured manner. This paper therefore provides a framework for Optimization with Constraint Learning (OCL) which we believe will help to formalize and direct the process of learning constraints from data. This framework includes the following steps: (i) setup of the conceptual optimization model, (ii) data gathering and preprocessing, (iii) selection and training of predictive models, (iv) resolution of the optimization model, and (v) verification and improvement of the optimization model. We then review the recent OCL literature in light of this framework, and highlight current trends, as well as areas for future research