The augmented lagrangian method can approximately solve convex optimization with least constraint violation

doi:10.1007/s10107-022-01843-2

CORC > 数学与系统科学研究院 > 中国科学院数学与系统科学研究院 > 计算数学与科学工程计算研究所

	The augmented lagrangian method can approximately solve convex optimization with least constraint violation
	Dai, Yu-Hong 2,3; Zhang, Liwei 1
刊名	MATHEMATICAL PROGRAMMING
	2022-06-17
页码	35
关键词	Convex optimization Least constraint violation Augmented Lagrangian method Shifted problem Optimal value mapping Solution mapping Dual function Conjugate dual
ISSN号	0025-5610
DOI	10.1007/s10107-022-01843-2
英文摘要	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 paper 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 constraint violation and numerical results verify our theoretical results.
WOS研究方向	Computer Science ; Operations Research & Management Science ; Mathematics
语种	英语
出版者	SPRINGER HEIDELBERG
WOS记录号	WOS:000812444200001
内容类型	期刊论文
源URL	[http://ir.amss.ac.cn/handle/2S8OKBNM/61538]
专题	计算数学与科学工程计算研究所
通讯作者	Dai, Yu-Hong
作者单位	1.Dalian Univ Technol, Sch Math Sci, Dalian 116024, Peoples R China 2.Chinese Acad Sci, AMSS, ICMSEC, LSEC, Beijing 100190, Peoples R China 3.Univ Chinese Acad Sci, Sch Math Sci, Beijing 100049, Peoples R China
推荐引用方式 GB/T 7714	Dai, Yu-Hong,Zhang, Liwei. The augmented lagrangian method can approximately solve convex optimization with least constraint violation[J]. MATHEMATICAL PROGRAMMING,2022:35.
APA	Dai, Yu-Hong,&Zhang, Liwei.(2022).The augmented lagrangian method can approximately solve convex optimization with least constraint violation.MATHEMATICAL PROGRAMMING,35.
MLA	Dai, Yu-Hong,et al."The augmented lagrangian method can approximately solve convex optimization with least constraint violation".MATHEMATICAL PROGRAMMING (2022):35.