学术报告报告题目: The Augmented Lagrangian Method Can Approximately Solve Convex Optimization with Least Constraint Violation
报 告 人: 张立卫 教授
报告时间: 2025年 11月 28 日（星期五）14:30-16:00
地 点：1号楼1303
邀 请 人: 潘少华教授
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数学学院
2025年11月25日
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. This is a joint work with Professor Yu-Hong Dai.
报告人简介：
张立卫，东北大学智能工业与系统优化国家级前沿科学研究中心教授。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”和“均衡优化”。他目前主持一项国家重点研发计划课题，完成和主持自然科学基金面上基金多项, 重点基金子课题两项。在国际顶级期刊Mathematical Programming, Operations Research, SIAM Journal on Optimization, Mathematics of Operations Research,Mathematics of Computation, JMLR, IEEE Transactions on Automatic Control 发表论文二十余篇。曾任中国运筹学会数学规划分会副理事长，中国运筹学会常务理事，中国运筹学会金融工程与金融风险管理分会副理事长。现任中国运筹学会监事,《JAPOR》,《NACO》和《运筹学学报》编委，2020年获得中国运筹学会运筹研究奖。