Variational Analysis: Theory and Applications

Wayne State University, Detroit MI

Investigators

Abstract

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.