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

我的研究项目主要是研究经济学、工程学、运筹学和管理学中出现的一些非常重要的问题的理论和算法。******(1)二层规划是两个优化问题的序列，其中上层问题的约束区域由下层问题的解集隐式确定。近年来，双层规划方法在越来越多的领域得到了应用。我计划研究约束条件较弱且可验证的最优性条件，并计划设计有效的算法来解决它们。特别是，我将重点关注某些具有某些结构的双层程序。******(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础性问题。它可以被看作是一个涉及不确定性的双层程序。委托代理问题处理的是在信息不完全和不对称的条件下，委托人雇佣代理人来追求委托人的利益，但代理人的行为对委托人来说是不可观察的。它侧重于设计一种激励方案，委托人通过该方案寻求激励代理人以对委托人有利的方式选择活动。我将尝试开发专门为这类问题设计的必要和充分最优性条件，使用经济学中合理的假设。我也将尝试找到有效的数值算法来解决委托代理问题。******(3) Stackelberg微分对策模型是一个两级问题，两级都是最优控制问题。近年来，为了对冲突和协调问题进行建模，它被用于管理科学中的应用，如供应链和营销渠道管理。对于这样的问题，我将尝试推导最优性条件并设计有效的算法。******(4)近年来，稀疏性惩罚优化在各个应用领域受到越来越多的关注。为了应对快速增长的数据集规模，最近的研究一直集中在解决这些问题的一阶方法上。特别是，人们已经认识到，非凸甚至非lipschitz惩罚诱导的解比凸对应的解更稀疏。利用变分分析的最新发展，我将尝试为这种非光滑和非凸问题的一阶必要最优性系统的误差界推导可验证的充分条件。这个条件是成功研究求解这些问题的各种一阶方法的收敛性和/或收敛率的关键。******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，并对整个世界产生影响。

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.