Theory and algorithms for solving bilevel optimization and other important nonsmooth and/or nonconvex optimization problems

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In my research, I apply variational analysis to the following four very important problems arising from Economics, Engineering, Operations Research and Management Science.(1) The bilevel program is a sequence of two optimization problems where the constraint region of the upper level problem is determined implicitly by the solution set to the lower level problem. (2) The principal-agent problem is a fundamental problem that frequently occurs in Economics, Management Science and Political Science. It treats the difficulties that arise under conditions of incomplete and asymmetric information when a principal hires an agent to pursue the principal's interests but the agent's action is unobservable. (3) The Stackelberg differential game model is a bilevel optimization problem where both levels are optimal control problems. In recent years, it has been used to model applications in Management Science such as supply chain management and marketing channels, specifically conflicts and coordination issues. (4) In Science, Social Sciences and Engineering, regression models have been playing a major role and the least squares estimator has been widely used. Optimal design of experiments is defined as finding designs such that one can get accurate information about the regression model or the regression parameter from its least squares estimator.These problems are all intrinsically nondifferentiable and nonconvex and hence very difficult to solve. Variational analysis is an extension of convex analysis to encompass a variety of nondifferentiable functions (convex or nonconvex) and mappings. Variational analysis provides a powerful tool to study the problems I propose to solve. The goal of this proposal is to develop theories and algorithms for solving these problems. While post-doctoral fellows and graduate students can concentrate on theory development, undergraduate students can do numerical experiments on algorithms and computations. I believe that my research will significantly advance our knowledge about the four proposed problems and that the success of my research will benefit Canada and impact the world at large.