An exact algorithm for linear optimization problem subject to max-product fuzzy relational inequalities with fuzzy constraints

Fuzzy relational inequalities with fuzzy constraints (FRI-FC) are the generalized form of fuzzy relational inequalities (FRI) in which fuzzy inequality replaces ordinary inequality in the constraints. Fuzzy constraints enable us to attain optimal points (called super-optima) that are better solutions than those resulted from the resolution of the similar problems with ordinary inequality constraints. This paper considers the linear objective function optimization with respect to max-product FRI-FC problems. It is proved that there is a set of optimization problems equivalent to the primal problem. Based on the algebraic structure of the primal problem and its equivalent forms, some simplification operations are presented to convert the main problem into a more simplified one. Finally, by some appropriate mathematical manipulations, the main problem is transformed into an optimization model whose constraints are linear. The proposed linearization method not only provides a super-optimum (that is better solution than ordinary feasible optimal solutions) but also finds the best super-optimum for the main problem. The current approach is compared with our previous work and some well-known heuristic algorithms by applying them to random test problems in different sizes.Comment: 29 pages, 8 figures, 7 table

Advances in Optimization and Nonlinear Analysis

The present book focuses on that part of calculus of variations, optimization, nonlinear analysis and related applications which combines tools and methods from partial differential equations with geometrical techniques. More precisely, this work is devoted to nonlinear problems coming from different areas, with particular reference to those introducing new techniques capable of solving a wide range of problems. The book is a valuable guide for researchers, engineers and students in the field of mathematics, operations research, optimal control science, artificial intelligence, management science and economics

Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition

Non-linear dependences in finance

The thesis is composed of three parts. Part I introduces the mathematical and statistical tools that are relevant for the study of dependences, as well as statistical tests of Goodness-of-fit for empirical probability distributions. I propose two extensions of usual tests when dependence is present in the sample data and when observations have a fat-tailed distribution. The financial content of the thesis starts in Part II. I present there my studies regarding the "cross-sectional" dependences among the time series of daily stock returns, i.e. the instantaneous forces that link several stocks together and make them behave somewhat collectively rather than purely independently. A calibration of a new factor model is presented here, together with a comparison to measurements on real data. Finally, Part III investigates the temporal dependences of single time series, using the same tools and measures of correlation. I propose two contributions to the study of the origin and description of "volatility clustering": one is a generalization of the ARCH-like feedback construction where the returns are self-exciting, and the other one is a more original description of self-dependences in terms of copulas. The latter can be formulated model-free and is not specific to financial time series. In fact, I also show here how concepts like recurrences, records, aftershocks and waiting times, that characterize the dynamics in a time series can be written in the unifying framework of the copula.Comment: PhD Thesi

The Fekete-Szego theorem with Local Rationality Conditions on Curves

Let $K$ be a number field or a function field in one variable over a finite field, and let $K^{sep}$ be a separable closure of $K$. Let $C/K$ be a smooth, complete, connected curve. We prove a strong theorem of Fekete-Szego type for adelic sets $E = \prod_v E_v$ on $C$, showing that under appropriate conditions there are infinitely many points in $C(K^{sep})$ whose conjugates all belong to $E_v$ at each place $v$ of $K$. We give several variants of the theorem, including two for Berkovich curves, and provide examples illustrating the theorem on the projective line, and on elliptic curves, Fermat curves, and modular curves

Nonlinear Analysis and Optimization with Applications

Nonlinear analysis has wide and significant applications in many areas of mathematics, including functional analysis, variational analysis, nonlinear optimization, convex analysis, nonlinear ordinary and partial differential equations, dynamical system theory, mathematical economics, game theory, signal processing, control theory, data mining, and so forth. Optimization problems have been intensively investigated, and various feasible methods in analyzing convergence of algorithms have been developed over the last half century. In this Special Issue, we will focus on the connection between nonlinear analysis and optimization as well as their applications to integrate basic science into the real world

Non linear dependences in finance

La thèse est composée de trois parties. La partie I introduit les outils mathématiques et statistiques appropriés pour l'étude des dépendances, ainsi que des tests statistiques d'adéquation pour des distributions de probabilité empiriques. Je propose deux extensions des tests usuels lorsque de la dépendance est présente dans les données, et lorsque la distribution des observations a des queues larges. Le contenu financier de la thèse commence à la partie II. J'y présente mes travaux concernant les dépendances transversales entre les séries chronologiques de rendements journaliers d'actions, c'est à dire les forces instantanées qui relient plusieurs actions entre elles et les fait se comporter collectivement plutôt qu'individuellement. Une calibration d un nouveau modèle à facteurs est présentée ici, avec une comparaison à des mesures sur des données réelles. Finalement, la partie III étudie les dépendances temporelles dans des séries chronologiques individuelles, en utilisant les mêmes outils et mesures de corrélations. Nous proposons ici deux contributions à l'étude du volatility clustering , de son origine et de sa description: l'une est une généralisation du mécanisme de rétro-action ARCH dans lequel les rendements sont auto-excitants, et l'autre est une description plus originale des auto-dépendances en termes de copule. Cette dernière peut être formulée sans modèle et n'est pas spécifique aux données financières. En fait, je montre ici aussi comment les concepts de récurrences, records, répliques et temps d'attente, qui caractérisent la dynamique dans les séries chronologiques, peuvent être écrits dans la cadre unifié des copules.The thesis is composed of three parts. Part I introduces the mathematical and statistical tools that are relevant for the study of dependences, as well as statistical tests of Goodness-of-fit for empirical probability distributions. I propose two extensions of usual tests when dependence is present in the sample data and when observations have a fat-tailed distribution. The financial content of the thesis starts in Part II. I present there my studies regarding the cross-sectional dependences among the time series of daily stock returns, i.e. the instantaneous forces that link several stocks together and make them behave somewhat collectively rather than purely independently. A calibration of a new factor model is presented here, together with a comparison to measurements on real data. Finally, Part III investigates the temporal dependences of single time series, using the same tools and measures of correlation. I propose two contributions to the study of the origin and description of volatility clustering : one is a generalization of the ARCH-like feedback construction where the returns are self-exciting, and the other one is a more original description of self-dependences in terms of copulas. The latter can be formulated model-free and is not specific to financial time series. In fact, I also show here how concepts like recurrences, records, aftershocks and waiting times, that characterize the dynamics in a time series can be written in the unifying framework of the copula.CHATENAY MALABRY-Ecole centrale (920192301) / SudocSudocFranceF

Fuzzy Sets, Fuzzy Logic and Their Applications 2020

The present book contains the 24 total articles accepted and published in the Special Issue “Fuzzy Sets, Fuzzy Logic and Their Applications, 2020” of the MDPI Mathematics journal, which covers a wide range of topics connected to the theory and applications of fuzzy sets and systems of fuzzy logic and their extensions/generalizations. These topics include, among others, elements from fuzzy graphs; fuzzy numbers; fuzzy equations; fuzzy linear spaces; intuitionistic fuzzy sets; soft sets; type-2 fuzzy sets, bipolar fuzzy sets, plithogenic sets, fuzzy decision making, fuzzy governance, fuzzy models in mathematics of finance, a philosophical treatise on the connection of the scientific reasoning with fuzzy logic, etc. It is hoped that the book will be interesting and useful for those working in the area of fuzzy sets, fuzzy systems and fuzzy logic, as well as for those with the proper mathematical background and willing to become familiar with recent advances in fuzzy mathematics, which has become prevalent in almost all sectors of the human life and activity

Optimization Models Using Fuzzy Sets and Possibility Theory

Optimization is of central concern to a number of disciplines. Operations Research and Decision Theory are often considered to be identical with optimization. But also in other areas such as engineering design, regional policy, logistics and many others, the search for optimal solutions is one of the prime goals. The methods and models which have been used over the last decades in these areas have primarily been "hard" or "crisp", i.e. the solutions were considered to be either feasible or unfeasible, either above a certain aspiration level or below. This dichotomous structure of methods very often forced the modeler to approximate real problem situations of the more-or-less type by yes-or-no-type models, the solutions of which might turn out not to be the solutions to the real problems. This is particularly true if the problem under consideration includes vaguely defined relationships, human evaluations, uncertainty due to inconsistent or incomplete evidence, if natural language has to be modeled or if state variables can only be described approximately. Until recently, everything which was not known with certainty, i.e. which was not known to be either true or false or which was not known to either happen with certainty or to be impossible to occur, was modeled by means of probabilities. This holds in particular for uncertainties concerning the occurrence of events. probability theory was used irrespective of whether its axioms (such as, for instance, the law of large numbers) were satisfied or not, or whether the "events" could really be described unequivocally and crisply. In the meantime one has become aware of the fact that uncertainties concerning the occurrence as well as concerning the description of events ought to be modeled in a much more differentiated way. New concepts and theories have been developed to do this: the theory of evidence, possibility theory, the theory of fuzzy sets have been advanced to a stage of remarkable maturity and have already been applied successfully in numerous cases and in many areas. Unluckily, the progress in these areas has been so fast in the last years that it has not been documented in a way which makes these results easily accessible and understandable for newcomers to these areas: text-books have not been able to keep up with the speed of new developments; edited volumes have been published which are very useful for specialists in these areas, but which are of very little use to nonspecialists because they assume too much of a background in fuzzy set theory. To a certain degree the same is true for the existing professional journals in the area of fuzzy set theory. Altogether this volume is a very important and appreciable contribution to the literature on fuzzy set theory