【作者】 袁艳红；

【导师】 张立卫；

【作者基本信息】 大连理工大学 ， 运筹学与控制论， 2012， 博士

【摘要】 广义Nash均衡问题(GNEP)产生于经济领域,由Arrow和Debreu于1954年正式提出。然而,目前关于广义Nash均衡问题的研究仍处于起步阶段。一部分研究是关于解的存在性,另一部分是关于问题求解的数值方法。较为有效的求解方法通常与变分不等式、半光滑问题、均衡问题和拟变分不等式问题相联系。本论文研究几个广义Nash均衡模型的数值方法,所取得的主要研究结果可概括如下：第二章主要讨论求解广义Nash均衡问题的惩罚方法。首先引入惩罚函数,在一定条件下证明了惩罚问题解的极限点就是原问题的解,且惩罚参数在有限次迭代后是一个有限常数。对于惩罚模型,运用光滑化Fischer-Burmeister函数将其Karush-Kuhn-Tucker系统转化为光滑方程组问题Ec=0,并在一定条件下证明了Ec在解点处Clarke广义微分的非奇异性。然后运用光滑牛顿法求解该光滑方程组,并给出了算法的全局收敛性和局部二次收敛性。最后给出数值算例,验证了算法的有效性。第三章主要讨论求解随机广义Nash均衡问题的惩罚函数方法。首先给出了随机广义Nash均衡问题的惩罚模型,并运用样本平均近似方法(SAA)得到其对应的SAA模型,证明了当样本容量趋于无穷时SAA模型的Karush-Kuhn-Tucker点列以概率1收敛到随机广义Nash均衡问题惩罚模型的Karush-Kuhn-Tucker点。然后,在一定条件下证明了SAA模型的Karush-Kuhn-Tucker系统在解点处的Clarke广义微分的非奇异性。最后给出了数值算例,说明基于SAA模型惩罚函数方法可以用来求解随机广义Nash均衡问题。第四章研究求解随机广义Nash均衡问题的光滑牛顿法。首先引入样本近似方法(SAA)得到原随机问题的SAA模型。对于样本容量为I的SAA模型,本章引入光滑化的Fischer-Burmeister函数将其Karush-Kuhn-Tucker系统转化为光滑方程组E1=0。然后,在一定条件下证明了E1在SAA解点处的Clarke广义微分的非奇异性。最后分析了光滑牛顿算法的全局收敛性和局部二次收敛性并给出了说明性的数值算例。第五章主要研究了求解二阶锥约束的广义Nash均衡问题的光滑牛顿法。首先用光滑化的投影函数将问题的Karush-Kuhn-Tucker系统转化为光滑方程组。然后在一定条件下证明了光滑方程组在解点处Clarke广义微分的非奇异性。最后用光滑牛顿法求解该光滑方程组,给出了算法的全局收敛性和局部二次收敛性。最后给出说明性数值算例。更多还原

【Abstract】 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.更多还原