Numerical Method For Solving Several Generalized Nash Equilibrium Problems

The generalized Nash equilibrium problems (GNEP), originated from economy, was formally proposed by Arrow and Debreu in1954. Howeverat present the study of GNEP is in its infancy. Some works are about the solution existence and others are about numerical methods. The efficient numerical methods depend upon recent development in variational inequalities, semi smooth equations, mathematical programs with equilibrium constraints and quasi-variational inequality problems. The dissertation focuses on the study of numerical methods for solving several generalized Nash equilibrium problems. The main results of this dissertation can be summarized as follows:Chapter2considers a penalty function method for solve the generalized Nash equilibrium problems. Firstly, we introduce the penalty function method and demonstrate that any accumulation point of the sequence of solutions for penalized problems is a solution to the GNEP and the penalty parameter becomes a constant after a finite iterations under some conditions. For the penalized model, the Karush-Kuhn-Tucker condition is formulated into a smoothing equation EC-0with the help of smoothing Fischer-Burmeister function. Under some conditions, we demonstrate the nonsingularity of Clarke's generalized Jacobian of EC at solution points. Furthermore, we give the global convergence and local quadratic rate of the smoothing Newton method. To the end, illustrative examples are given to show the efficiency of the penalty function method.Chapter3mainly discuses the penalty function method for solving stochastic generalized Nash equilibrium problems. Firstly, we give the penalty model of the stochastic generalized Nash equilibrium problem and the SAA model of the penalty model. We demonstrate that the sequence of Karush-Kuhn-Tucker points of the SAA model converges to a Karush-Kuhn-Tucker point of the true problem with probability one at an exponential rate when the sample size tends to infinity. Under some conditions, for the SAA model we prove the nonsingularity of Clarke's generalized Jacobian of the Karush-Kuhn-Tucker system at the solution points. To the end, some examples are given to show that the penalty method based on SAA method is practical to solve the stochastic generalized Nash equilibrium problem. Chapter4discusses the smoothing Newton method for solving the stochastic generalized Nash equilibrium problems. Firstly, we formulate the stochastic generalized Nash equilibrium problem into a SAA model. For the SAA model with sample size I, we introduce the smoothing Fischer-Burmeister function to transform the Karush-Kuhn-Tucker system into a smoothing equation E1=0. Under some conditions, we prove that the Clarke's generalized Jacobian of E1at the solution points is nonsingular. Furthermore, the global convergence and local quadratic rate of the smoothing Newton method are given and illustrative examples are given, too.Chapter5deals with the second-order cone constrained generalized Nash equilibrium problems by using the smoothing Newton method. Firstly, the Karush-Kuhn-Tucker system of the second-order cone constrained generalized Nash equilibrium problem is formulated into a smoothing equation with help of the smoothing metric projectors. Under some conditions, we prove that the Clarke's generalized Jacobian of the smoothing equation in solution points is nonsingular. To the end, we present global convergence and local quadratic rate of the smoothing Newton method. Finally, several illustrative examples are given.