Equilibrium problems provide a mathematical framework which includes optimization, variational inequalities, fixed-point and saddle point problems, and noncooperative games as particular cases. This general format received an increasing interest in the last decade mainly because many theoretical and algorithmic results developed for one of these models can be often extended to the others through the unifying language provided by this common format. This survey paper aims at covering the main results concerning the existence of equilibria and the solution methods for finding them
v vi Preface able: as optimization fits in this format, nonlinear programming techniques have often been the key tool of their work. The book aims at addressing in particular two core issues such as the existence and computation of equilibria. The first chapter illustrates a sample of applications, the second addresses the main theoretical issues, while the third introduces the main algorithms available for computing equilibria. A final chapter is devoted to quasi-equilibria, a more general format that is needed to cover more complex applications having additional features such as shared resources in noncooperative games. Finally, basic material on sets, functions and multivalued maps that are exploited throughout the book are summarized in the appendix. To make the book as readable as possible, examples and applications have been included. We hope that this book may serve as a basis for a second level academic course or a specialised course in a Ph.D. programme and stimulate further interest in equilibrium problems.
A globally convergent algorithm for equilibrium problems with differentiable bifunctions is proposed. The algorithm is based on descent directions of a suitable family of gap functions. The novelty of the approach is that assumptions which guarantee that the stationary points of the gap functions are global optima are not required
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