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

doi:10.1007/s10107-022-01843-2

	The augmented lagrangian method can approximately solve convex optimization with least constraint violation
	Dai, Yu-Hong1,2 ; Zhang, Liwei 3
	2022-06-17
发表期刊	MATHEMATICAL PROGRAMMING
ISSN	0025-5610
页码	35
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.
关键词	Convex optimization Least constraint violation Augmented Lagrangian method Shifted problem Optimal value mapping Solution mapping Dual function Conjugate dual
DOI	10.1007/s10107-022-01843-2
收录类别	SCI
语种	英语
WOS研究方向	Computer Science ; Operations Research & Management Science ; Mathematics
WOS类目	Computer Science, Software Engineering ; Operations Research & Management Science ; Mathematics, Applied
WOS记录号	WOS:000812444200001
出版者	SPRINGER HEIDELBERG
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/61539
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.Chinese Acad Sci, AMSS, ICMSEC, LSEC, Beijing 100190, Peoples R China 2.Univ Chinese Acad Sci, Sch Math Sci, Beijing 100049, Peoples R China 3.Dalian Univ Technol, Sch Math Sci, Dalian 116024, 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.