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| 求解广义纳什均衡问题的极小极大方法 |
| A minimax approach for solving the generalized Nash equilibrium problem |
| 投稿时间：2013-07-11 修订日期：2013-09-13 |
| DOI： |
| 中文关键词: 纳什均衡 广义纳什均衡 变分不等式 半光滑牛顿法 |
| 英文关键词: Nash equilibrium problem generalized Nash equilibrium problem variational inequality problem semismooth Newton method |
| 基金项目:国家自然科学基金资助项目(11071029 , 91130007) |
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| 摘要点击次数: 1947 |
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| 中文摘要: |
| 广义纳什均衡问题是较经典纳什均衡问题更一般的优化问题, 这类问题在很多领域中都有重要的应用, 然而对广义纳什均衡问题数值方法的研究却远远不如对经典纳什均衡问题研究的那样完善. 本文利用正则化Nikaido-Isoda函数将一类广义纳什均衡问题的求解转化为一个极小极大问题的求解. 利用Fischer-Burmeister函数将与极小极大问题的必要性条件等价的变分不等式的Karush-Kuhn-Tucker系统转化为一个半光滑方程组. 应用牛顿法求解此方程组, 并给出了半光滑牛顿法局部超线性收敛的充分性条件. 数值结果验证了极小极大方法的有效性. |
| 英文摘要: |
| The generalized Nash equilibrium problem (GNEP) is a generalization of the standard Nash equilibrium problem. This class of problems find a lot of applications in many fields. However, compared with the classical Nash equilibrium problem, the numerical solution algorithms for GNEPs are extremely scare in the literature. Using the regularized Nikaido-Isoda function, the generalized Nash equilibrium problem is reformulated as a minimax problem. Based on the Fischer-Burmeister function, the Karush-Kuhn-Tucker system of the variational inequality problem equivalent to the necessary conditions for this minimax problem, is transformed into a semismooth system of equations. The semismooth Newton method is used to solve the equation and sufficient conditions for the local superlinear convergence of the semismooth Newton method are derived. Numerical results show that the minimax approach for solving the generalized Nash equilibrium problem is practical. |
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