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

Abstract. Decentralised smart contracts represent the next step in the development of protocols that support the interaction of independent players without the presence of a coercing authority. Based on protocols a la BitCoin for digital currencies, smart contracts are believed to be a potentially enabling technology for a wealth of future applications. The validation of such an early developing technology is as necessary as it is complex. In this paper we combine game theory and formal models to tackle the new challenges posed by the validation of such systems.

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

Second order optimality conditions for differentiable multiobjective problems RAIRO. Recherche opérationnelle, tome 34, n o 4 (2000), p. 411-426 © AFCET, 2000, tous droits réservés. L'accès aux archives de la revue « RAIRO. Recherche opérationnelle » implique l'accord avec les conditions générales d'utilisation (http://www. numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ RAIRO Opérations Research RAIRO Oper. Res. 34 (2000) 411-426 SECOND ORDER OPTIMALITY CONDITIONS FOR DIFFERENTIABLE MULTIOBJECTIVE PROBLEMS (*) by Giancarlo BIGI (*) and Marco CASTELLANÏ ( 2 )Communicated by Franco GIANNESSI Abstract. -A second order optimality condition for multiobjective optimization with a set constraint is developed; this condition is expressed as the impossibility of nonhomogeneous linear Systems. When the constraint is given in terms of inequalities and equalities, it can be turned into a John type multipliers rule, using a nonhomo geneous Motzkin Theorem of the Alternative. Adding weak second order regularity assumptions, Karush, Kuhn-Tucker type conditions are therefore deduced.