Nonlinear Programming Techniques for Equilibria

Abstract

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 Dynamical System for Strongly Pseudo-monotone Equilibrium Problems

“…Indeed, a saddle point of G is a Nash equilibrium in a two-person zero-sum game, that is a noncooperative game where the cost function of the first player is G and the cost function of the second player is −G (see, e.g., [2,8]. )…”

“…In recent years, a large number of applications have been described successfully via the concept of equilibrium solution and therefore many researchers devoted their efforts to study EPs. For an excellent survey on existence of equilibrium points and solution methods for finding them, we refer the readers to the new monograph [8].…”

“…Since EP is a general model, the methods for solving particular problems can be often extended to EPs; see for example, [8][9][10][11][12][13][14][15][16][17][18][19]. Among them, fixed point-type methods play an important role, because they are simple in form and useful in practice.…”

Newton’s Method and Differential Dynamic Programming for Unconstrained Nonlinear Dynamic Games

“…GNEPs are reformulated to a set of necessary conditions via KKT conditions, which is in the form of variational inequalities (VI). These inequalities can be solved via generic VI methods or classic feasibility problem methods, such as Newton's method [20] or others [45]. In particular, Newton's method converts the complementarity conditions to equality constraints via complementarity functions.…”

A splitting subgradient algorithm for solving equilibrium problems involving the sum of two bifunctions and application to cournot-nash model

“…Many papers concerning the solution existence, stabilities as well as algorithms for Problem (EP ) have been published (see e.g. [10,13,16,20,25,26,27], the survey paper [4], and the interesting monograph [5]). A basic method for Problem (EP ) is the subgradient (or projection) one, where the sequence of iterates is defined by taking…”