学术报告

The Augmented Lagrangian Method Can Approximately Solve Convex Optimization with Least Constraint Violation

张立卫

（大连理工大学）

报告时间: 2022年5月17日 星期三  15：00-16：00

报告地点: 沙河主楼E405

腾讯会议：861-172-286

报告摘要: There are many important practical optimization problems whose feasible regions are not known to be nonempty or not, and optimizers of the objective function with the least constraint violation prefer to be found. A natural way for dealing with these problems is to extend the nonlinear optimization problem as the one optimizing the objective function over the set of points with the least constraint violation. This leads to the study of the  shifted problem. This report focuses on the constrained convex optimization problem. The sufficient condition for the closedness of the set of feasible shifts is presented and the continuity properties of the optimal value function and the solution mapping for the shifted problem are studied. Properties of the conjugate dual of the shifted problem are discussed through the relations between the dual function and the optimal value function. The solvability of the dual of the optimization problem with the least constraint violation is investigated. It is shown that, if the least violated shift is in the domain of the subdifferential of the optimal value function, then this dual problem has an unbounded solution set. Under this condition, the optimality conditions for the problem with the least constraint violation are established in term of the augmented Lagrangian. It is shown that the augmented Lagrangian method has the properties  that the sequence of shifts converges to the least violated shift and  the sequence of multipliers is  unbounded. Moreover, it is proved that the augmented Lagrangian method is able to find an approximate solution to the problem with the least constraint violation and it has linear rate of convergence under an error bound condition. The augmented Lagrangian method is applied to an illustrative convex second-order cone constrained optimization problem with least violation constraint and numerical results verify the theoretical results obtained in this paper. This is a joint work with Professor Yu-Hong Dai.

报告人简介：

张立卫教授，大连理工大学154.com皇冠运筹学与控制论业博士生指导教师，金融数学与保险精算专业博士生指导教师。他于1989年，1992年，1998年分别在大连理工大学获得理学学士，硕士，博士学位，1999-2001在中科院计算数学所从事博士后工作。目前的研究兴趣是“矩阵优化”，“随机规划”与“均衡优化”。目前主持国家重点研发计划课题一项，主持国家自然科学基金面上项目一项，完成两项国家自然科学基金重点项目子课题，完成国家自然科学基金面上项目多项项。在国际顶级期刊Math. Programming, Operations Research, SIAM J. Optimization, Mathematics of Operations Research, Mathematics of Computation 发表论文10多篇，2020年获得中国运筹学会运筹研究奖，现任中国运筹学会常务理事，中国运筹学会数学规划分会副理事长，中国运筹学会金融工程与金融风险管理分会副理事长,《JAPOR》和《运筹学学报》编委。

邀请人：韩德仁