参数优化问题解映射的Lipschitz性质和广义可微性研究

The study on stability and differentiability for parametric optimization problems is an important research area of optimization theory. Recently，many scholars more and more follow with interest the Lipschitz property and generalized differentiability of set-valued solution mappings for the problems, however,the conditions added frequently involve the solution information,or although the solution mappings of vector optimization problems are set-valued,yet when being applied the corresponding scalar optimization problems,the solution mappings are single-valued. Being planned to adopt the tools and methods in set-valued and variational analysis,and on condition that the solution mappings are set-valued, the project shall study the Lipschitz property,Robinson's metric regularity and generalized differentiability of solution mappings for parametric optimization problems, and shall reveal the importance of gap functions in parametric vector optimization problems, and shall seek the relationships between Robinson's metric regularity and generalized differentiability for the solution mappings, in addition, shall apply the conclusion obtained to traffic network equilibrium problems and game equilibrium problems.The research contents of this project have important theoretical significances and practical values, and they are important inheritances and developments for set-valued and variational analysis theory.

参数优化问题的稳定性和可微性研究是最优化理论的一个重要的研究方向。最近，许多学者越来越关注该类问题集值解映射的Lipschitz性质和广义可微性，但往往所给条件涉及到解集的信息或者虽然针对向量问题解映射是集值的而应用到相应的标量问题时解映射却变为单值的。本课题拟采用集值分析和变分分析的工具和方法，在保证解映射是集值的前提下，研究参数优化问题的解映射的Lipschitz性质、Robinson度量正则和广义可微性，揭示间隙函数在参数向量优化问题中的重要作用，寻求解映射的Robinson度量正则和广义可微性之间的蕴含关系，并将所得结论应用到交通网络平衡问题和博弈均衡中。本课题的研究内容具有重要的理论价值和应用前景，是对集值分析和变分分析理论的重要继承和发展。

参数优化问题的稳定性和广义可微性是最优化理论的一个重要研究方向。本课题采用集值分析和变分分析的工具和方法，在保证解映射是集值的前提下，研究参数优化问题解映射的Lipschitz性质和广义可微性。首先，本课题利用非线性标量化方法，在充分挖掘一类非线性标量化函数的性质的前提下，分别得到了参数向量平衡问题和广义参数向量拟平衡问题解映射的Holder连续性，推广和改进了已有的一些结果。其次，本课题利用变分不等式问题的间隙函数的性质，得到了该问题的误差界，并将此结果应用到该问题的一个下降算法的收敛性中，同时应用到参数变分不等式问题解映射的Holder连续性和广义可微性中。再次，将各类问题解映射转化为广义方程，利用广义方程的研究方法和手段，得到了一类隐函数的Lipschitz连续性和图像导数表达式。最后，研究了一类正则和Lipschitz函数的误差界的模，给出了模的上界和下界的刻画，并分别研究了上界和下界达到的一些充分条件，而且在一些特殊情况下还给出了几何上的解释。