Variational Analysis: Theory and Applications

1.所提出的研究主要涉及发展无限维空间中的一阶广义变分分析理论，有限维中二阶变分分析的一些基本方面，以及它们在优化、控制、力学和经济学中的重要问题上的应用。广义微分的主要工具是基于非光滑物体的法锥、次微分和余导数，而PI在这方面的研究由来已久。该项目将特别关注二阶次微分理论及其在变分系统的Lipschitzian稳定性、带平衡约束的数学规划以及连续介质力学的一些问题中的应用。它将包含在控制系统的动态优化领域的新发展和应用，这些控制系统由微分方程组、时滞微分方程组和偏微分方程式/包含所支配。它还旨在开发变分分析在研究福利经济学均衡模型中的帕累托最优性方面的新应用。2.变分分析被认为是现代应用数学中一个卓有成效的领域，它一方面致力于在具有复杂的非光滑约束的大规模数学模型中寻找最优解，另一方面开发优化策略和技术来解决应用科学、控制、经济、工程、力学等各个领域中出现的广泛的现实问题。特别是，所提出的极小极大控制设计方法在很大程度上是由环境系统的应用所推动的。平衡约束最优化问题的变分稳定性研究是基于连续介质力学中材料设计的实际问题而提出的。变分分析的新应用将发展到福利经济学的多目标模型。

1. The proposed research mainly deals with developing the first-order generalized differential theory of variational analysis in broad classes of infinite-dimensional spaces, with some basic aspects of the second-order variational analysis in finite dimensions, and with their applications to important problems arising in optimization, control, mechanics, and economics. The main tools of the generalized differentiation in the proposed research are based on the normal cones, subdifferentials, and coderivatives of nonsmooth objects, in the study of which the PI has been involved for a long time. The project will pay a particular attention to the second-order subdifferential theory and its applications to the Lipschitzian stability of variational systems, mathematical programs with equilibrium constraints, and to some problems of continuum mechanics. It will contain new development and applications in the area of dynamic optimization for control systems governed by differential, delay-differential, and partial differential equations/inclusions. It also aims to develop new applications of variational analysis to the study of Pareto optimality in equilibrium models of welfare economics. 2. Variational analysis has been recognized as a fruitful area in the modern Applied Mathematics that, on one hand, is concerned with finding the best solutions in large-scale mathematical models with complicated ``nonsmooth" constraints and, on the other hand, develops optimization strategies and technologies to solve a broad spectrum of real-life problems arising in various areas of applied science, control, economics, engineering, mechanics, etc. The proposed research aims to develop new methods of variational analysis and its application to some important problems in optimal control, economics, and mechanics. In particular, the proposed methods of minimax control design are largely motivated by applications to environmental systems. The proposed study of variational stability for optimization problems with equilibrium constraints is motivated by applications to practical problems of material design in continuum mechanics. New applications of variational analysis will be developed to multiobjective models of welfare economics.