Generalized Nash equilibrium problems have become very important as a modeling tool during the last decades. The aim of this survey paper is twofold. It summarizes recent advances in the research on computational methods for generalized Nash equilibrium problems and points out current challenges. The focus of this survey is on algorithms and their convergence properties. Therefore, we also present reformulations of the generalized Nash equilibrium problem, results on error bounds and properties of the solution set of the equilibrium problems.
In this paper, we first derive a characterization of the solution set of a continuously differentiable system of equations subject to a closed feasible set. Assuming that a constrained local error bound condition is satisfied, we prove that the solution set can locally be written as the intersection of a differentiable manifold with the feasible set. Based on the derivation of this result, we then show that the projected Levenberg-Marquardt method converges locally R-linearly to a possibly nonisolated solution under significantly weaker conditions than previously done.
ARTICLE HISTORY
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On the Globalization of Wilson-type Optimization Methods by Means of Generalized Reduced GradientMethods. For solving nonlinear optimization problems, i.e. for the determination of Kuhn-Tucker points a numerical method is proposed. The considerations continue investigations of Best/ Br~iuninger/Ritter/Robinson and Kleinmichel/Richter/Sch6nefeld. In these papers (published in this journal) different local methods are combined with a penalty method in such a way that global convergence can be guaranteed. In order to show that the basic principle of coupling is applicable to a number of further globally convergent methods a local Wilson-type method is now initialized by a feasible direction method that uses reduced gradients. In both phases of the method similar subproblems (special quadratic programs) occur. Therefore, in contrast to the papers mentioned above systems of linear equations have to be solved exclusively. Under usual assumptions the algorithm is shown to be globally and superlinearly convergent.AMS Subject Classifications: 90C30, 65K05.
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