Research on Variational Analysis and Its Applications

近年来，现代变分分析取得了重大进展，已成为数学中一个富有成果的领域。该项目主要关注先进的变分原理和技术的发展，并将其广泛应用于各种优化与平衡问题（包括双层和半无限规划，平衡约束下的优化与平衡问题，多目标优化）；数值算法的适定性和条件；常微分、函数微分、偏微分等系统的最优控制。由于非光滑函数和集值映射在应用高级变分原理和技术时自然而频繁地出现，因此该项目强烈地解决了此类非标准对象的广义微分理论。本项目开发的优化/变分原理和技术主要旨在应用于经济学、环境科学、电力市场、工程和力学中出现的现实问题。通过以适合应用先进变分技术的形式对这些问题进行建模，并为他们的研究发展理论和数值方法，该项目将解决上述领域中出现的一些具有实际重要性的现实问题。

Recent years have witnessed a significant progress in modern variational analysis, which has become a fruitful area of mathematics with a strong emphasis on applications. The project mainly concerns the development of advanced variational principles and techniques with their broad applications to various problems in optimization and equilibria (including bilevel and semi-infinite programming, optimization and equilibrium problems with equilibrium constraints, multiobjective optimization); well-posedness and conditioning of numerical algorithms; optimal control of ordinary differential, functional differential, and partial differential systems, etc. Since nonsmooth functions and set-valued mappings appear naturally and frequently in applying advanced variational principles and techniques, the project strongly addresses generalized differential theory for such nonstandard objects.Optimization/variational principles and techniques developed in this project primarily aim at applications to real-life problems arising in economics, environmental science, electricity markets, engineering, and mechanics. By modeling such problems in the forms suitable for applications of advanced variational techniques and developing theoretical and numerical methods for their study, the project will solve a number of real-life problems of practical importance arising in the aforementioned areas.