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

在我的研究中，我将变分分析应用于经济学、工程学、运筹学和管理学中出现的以下四个非常重要的问题。(1)双层规划是由两个优化问题组成的序列，其中上层问题的约束区域由下层问题的解集隐式确定。(2)委托代理问题是经济学、管理学和政治学中经常出现的一个基本问题。它处理在信息不完全和不对称的条件下，当委托人雇佣代理人来追求委托人的利益，但代理人的行为是不可观察的时所产生的困难。(3) Stackelberg微分对策模型是一个两级优化问题，两级都是最优控制问题。近年来，它已被用于模拟管理科学中的应用，如供应链管理和营销渠道，特别是冲突和协调问题。(4)在科学、社会科学和工程领域，回归模型发挥了重要作用，最小二乘估计量得到了广泛的应用。实验的最优设计被定义为找到这样的设计，即人们可以从回归模型或回归参数的最小二乘估计中获得准确的信息。这些问题本质上都是不可微且非凸的，因此很难解决。变分分析是凸分析的扩展，包括各种不可微函数（凸或非凸）和映射。变分分析为研究我提出要解决的问题提供了一个强有力的工具。本提案的目标是发展解决这些问题的理论和算法。博士后和研究生可以专注于理论发展，本科生可以做算法和计算方面的数值实验。我相信我的研究将大大提高我们对这四个问题的认识，我的研究的成功将使加拿大受益，并在很大程度上影响世界。

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.