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

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My program of research focuses on studying theories and algorithms for solving some very important problems arising in 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. Recently, the bilevel programming methodology has been applied to more and more areas. I plan to study optimality conditions with weaker and verifiable constraint qualifications, and plan to design efficient algorithms for solving them. In particular, I will focus on certain bilevel programs with some structures. (2) The principal-agent problem is a fundamental problem that frequently occurs in economics, management science and political science. It can be viewed as a bilevel program involving uncertainty. The principal-agent problem 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 actions are unobservable to the principal. It focuses on designing an incentive scheme with which the principal seeks to motivate the agent to choose activities in a manner advantageous to the principal. I will try to develop necessary and sufficient optimality conditions specially designed for such problems using assumptions which are reasonable in Economics. I will also try to find efficient numerical algorithms for solving the principal-agent problem. (3) The Stackelberg differential game model is a bilevel problem where both levels are optimal control problems. In recent years, it has been used to model the applications in management science such as the supply chain and marketing channels management in order to model conflicts and coordination issues. For such a problem, I will try to derive optimality conditions and design efficient algorithms. (4) In recent years, optimization with sparsity-inducing penalties has received increasing attention in various application areas. To cope with the rapidly growing size of datasets, recent research has been focusing on first-order methods for solving these problems. In particular, it has been recognized that a non-convex or even non-Lipschitz penalty induces sparser solutions than convex counterparts. Using recent developments in variational analysis, I will try to derive verifiable sufficient conditions for error bounds for the first order necessary optimality system for such a nonsmooth and nonconvex problem. Such a condition is key to the successful study of the convergence and/or the convergence rate of various first order methods for solving these problems.I believe that my research will significantly advance our knowledge of the four proposed problems, and that the success of my research will benefit Canada and impact the world at large.