Variational Analysis: Theory and Applications

1. 本研究主要涉及在无限维空间中发展一阶变分分析的广义微分理论，在有限维空间中发展二阶变分分析的一些基本方面，以及它们在优化、控制、力学和经济等领域的应用。本文研究的广义微分的主要工具是基于非光滑物体的正锥、次微分和协导数，PI在这方面的研究已经进行了很长时间。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、平衡约束的数学程序和连续介质力学的一些问题中的应用。它将包含新的发展和应用领域的动态优化控制系统由微分，延迟微分和偏微分方程/包含。本研究也旨在发展变分分析在福利经济学均衡模型中帕累托最优性研究的新应用。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，一方面，它涉及在具有复杂“非光滑”约束的大规模数学模型中寻找最佳解，另一方面，它开发了优化策略和技术，以解决应用科学、控制、经济学、工程、力学等各个领域中出现的广泛的现实问题。本研究旨在发展变分分析的新方法，并将其应用于最优控制、经济学和力学中的一些重要问题。特别是，所提出的极大极小控制设计方法在很大程度上是由环境系统的应用所驱动的。本文提出的平衡约束优化问题的变分稳定性研究是由连续介质力学中材料设计实际问题的应用所激发的。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.