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This book presents powerful techniques for solving global optimization problems on manifolds by means of evolutionary algorithms, and shows in practice how these techniques can be applied to solve real-world problems. It describes recent findings and well-known key facts in general and differential topology, revisiting them all in the context of application to current optimization problems. Special emphasis is put on game theory problems. Here, these problems are reformulated as constrained global optimization tasks and solved with the help of Fuzzy ASA. In addition, more abstract examples, including minimizations of well-known functions, are also included. Although the Fuzzy ASA approach has been chosen as the main optimizing paradigm, the book suggests that other metaheuristic methods could be used as well. Some of them are introduced, together with their advantages and disadvantages.

Readers should possess some knowledge of linear algebra, and of basic concepts of numerical analysis and probability theory. Many necessary definitions and fundamental results are provided, with the formal mathematical requirements limited to a minimum, while the focus is kept firmly on continuous problems. The book offers a valuable resource for students, researchers and practitioners. It is suitable for university courses on optimization and for self-study.



Introductory Information


Chapter 1. The Many Aspects of Global Optimization

This chapter contains several considerations about global optimization methods, exposing in a condensed way several of their main qualitative characteristics. Taking into account that the focus in this book will be on the combination of an evolutionary method along with results of Topology and corresponding applications, we’ll start to pave the way that will take us to the practical utilization of that approach, and stochastic optimization methods, in general.
Hime Aguiar e Oliveira Junior

Chapter 2. Overview of Current Metaheuristic Paradigms

At this point some of the most visible paradigms related to global optimization using metaheuristics are described. The general intention is to describe well-established methods and show their usefulness in several difficult optimization scenarios. Although there exist a reasonable number of effective algorithms in this class, only a few of them are presented here, in the hope that the selection may give the reader a good idea of the whole context. In principle each one may be used as an “optimization engine” when doing global optimization on manifolds, to be described later.
Hime Aguiar e Oliveira Junior

Global Optimization on Manifolds


Chapter 3. Evolutionary Global Optimization on Manifolds

In this chapter it is described an approach to globally optimize real valued functions defined on topological manifolds. The functions under investigation do not need to be differentiable or even continuous. It is shown that optimization processes may take place so that candidate points remain restricted to the manifolds that contain their domains—the evolution occurs inside them during the entire optimization process. The presented paradigm is adequate for use with virtually all already existing metaheuristics, but here the algorithm known as Fuzzy Adaptive Simulated Annealing (Fuzzy ASA) is used in order to exemplify the overall scheme. After exposing the fundamental ideas, some examples will illustrate the efficacy of the proposed method.
Hime Aguiar e Oliveira Junior

Chapter 4. Constrained Global Optimization on Manifolds

This chapter presents techniques for dealing with constrained global optimization of real valued functions defined on smooth manifolds, subject to equality constraints. Functional constraints must satisfy certain smoothness conditions, not excluding simultaneous restrictions of different types, being the effect of dimensional reduction proportional to the number of equality restrictions. The problems under study do not need to restrict cost functions to be differentiable or even continuous, and the optimization task is done so as to keep candidate points inside corresponding submanifolds, evolving therein along the optimization process. The techniques may be employed together with an extensive family of already tested evolutionary methods and, after introducing the fundamental ideas, selected examples will demonstrate the effectiveness of the presented methods.
Hime Aguiar e Oliveira Junior

Further Applications of Fuzzy ASA


Chapter 5. Nash Equilibria of Finite Strategic Games and Fuzzy ASA

In this chapter various significant results obtained by means of the application of the Fuzzy Adaptive Simulated Annealing (Fuzzy ASA) algorithm are introduced—the aim is to find all Nash equilibria of finite normal form games. To get there, Fuzzy ASA has been modified in order to incorporate techniques, based on space-filling curves, able to find adequate starting points—several well-known strategic games are used to test the efficacy of the method. The obtained results are compared to previously published results that used similar techniques in order to solve the same problem but could not find all equilibria in all tests. As it is very important to study and model the interactions between agents, the Nash equilibrium concept is widely recognized as a powerful tool, adequate to discover situations in which joint strategies are optimal in the sense that players cannot benefit from changing unilaterally their strategies. In this fashion, any technique that may represent a true advancement, in terms of efficacy when finding whole sets of solutions for a given strategic game, is worth to invest in.
Hime Aguiar e Oliveira Junior

Chapter 6. Generalized Nash Equilibrium Problems and Fuzzy ASA

As an extension of the standard Nash equilibrium concept, the generalized Nash equilibrium (GNE) makes it possible to model and solve more general problems in several scenarios. Its most prominent advantage resides in that the GNE concept allows objective functions and constraints associated to each player to depend on the strategies of other agents, creating a more realistic environment. By studying GNE properties, problems in several fields, including Engineering and Economics, may be modelled and solved in an easier way. In this chapter a solution algorithm based on the Fuzzy ASA algorithm is introduced , evidencing that it is possible to transform many complex tasks into constrained global optimization problems—as such, they can be solved, in principle, by any effective global optimization algorithm, but here the main tool is Fuzzy ASA. The intention is to show that the presented approach may offer a simpler alternative for solving this type of problem in a less limited way, that is, not imposing strong conditions on the defining functions. After the theoretical explanation, many examples are presented in order to demonstrate the efficacy of the method.
Hime Aguiar e Oliveira Junior

Chapter 7. Studying Coalitions

A deep study assessing the feasibility of coalition formation in electric energy auctions is presented. A stochastic global optimization algorithm, when applied to the calculation of Nash–Cournot equilibria in several scenarios, makes it possible to obtain quantitative results concerning the profitability of coalition formation processes in diverse environments. Auxiliary Nash equilibrium problems are solved by transforming the original problem into a global optimization one and constructing cost functions which translate the associated constraints into mathematical relations, reflecting the benefit maximization trend of typical energy conversion and transmission firms. It is also indicated how to use the algorithm to estimate coupled constraint equilibria occurring when restrictions are imposed to businesses or marketplaces. In addition, the suggested method computes players’ payoffs in many configurations, comparing their profits and production levels under different market elasticities. Furthermore, solutions are based on cooperative game theory concepts, such as the bilateral Shapley value. It is shown that the adequacy of creating certain coalition configurations depends critically on demand \(\times \) price elasticity relationships. A case study based on the IEEE 30-bus system is used, for the sake of presenting and discussing in detail the paradigm. The presented method is far-reaching and uses the solution of generalized Nash equilibrium problems to obtain numerical data that will take us to the final decisions. As seen in the previous chapter, generalized Nash equilibrium problems address extensions of the well-known standard Nash equilibrium concept, making it possible to model and study more general configurations. As said before, GNEP’s have a larger scope, considering that they allow both objective functions and constraints of each player to depend on the strategies of other players. As can be observed from the literature, the study of such problems finds endless applications in several areas, including Medicine, Engineering, and Management Science, for example.
Hime Aguiar e Oliveira Junior

Chapter 8. Epilogue

This short chapter aims to summarize the content and intentions of the book.
Hime Aguiar e Oliveira Junior


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