A NEW FIRST-ORDER ALGORITHMIC FRAMEWORK FOR OPTIMIZATION PROBLEMS WITH ORTHOGONALITY CONSTRAINTS

doi:10.1137/16M1098759

	A NEW FIRST-ORDER ALGORITHMIC FRAMEWORK FOR OPTIMIZATION PROBLEMS WITH ORTHOGONALITY CONSTRAINTS
	Gao, Bin 1,2; Liu, Xin1,2 ; Chen, Xiaojun 3; Yuan, Ya-Xiang1
	2018
发表期刊	SIAM JOURNAL ON OPTIMIZATION
ISSN	1052-6234
卷号	28 期号:1 页码:302-332
摘要	In this paper, we consider a class of optimization problems with orthogonality constraints, the feasible region of which is called the Stiefel manifold. Our new framework combines a function value reduction step with a correction step. Different from the existing approaches, the function value reduction step of our algorithmic framework searches along the standard Euclidean descent directions instead of the vectors in the tangent space of the Stiefel manifold, and the correction step further reduces the function value and guarantees a symmetric dual variable at the same time. We construct two types of algorithms based on this new framework. The first type is based on gradient reduction including the gradient reflection (GR) and the gradient projection (GP) algorithms. The other one adopts a columnwise block coordinate descent (CBCD) scheme with a novel idea for solving the corresponding CBCD subproblem inexactly. We prove that both GR/GP with a fixed step size and CBCD belong to our algorithmic framework, and any clustering point of the iterates generated by the proposed framework is a first-order stationary point. Preliminary experiments illustrate that our new framework is of great potential.
关键词	orthogonality constraint Stiefel manifold Householder transformation gradient projection block coordinate descent
DOI	10.1137/16M1098759
语种	英语
资助项目	NSFC[11622112] ; NSFC[11471325] ; NSFC[91530204] ; NSFC[11688101] ; NSFC[11331012] ; NSFC[11461161005] ; National Center for Mathematics and Interdisciplinary Sciences, CAS ; Key Research Program of Frontier Sciences, CAS ; Hong Kong Research Council[N_PolyU504/14]
WOS研究方向	Mathematics
WOS类目	Mathematics, Applied
WOS记录号	WOS:000424527900012
出版者	SIAM PUBLICATIONS
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/29680
专题	计算数学与科学工程计算研究所
通讯作者	Gao, Bin
作者单位	1.Chinese Acad Sci, Acad Math & Syst Sci, State Key Lab Sci & Engn Comp, Beijing, Peoples R China 2.Univ Chinese Acad Sci, Beijing, Peoples R China 3.Hong Kong Polytech Univ, Dept Appl Math, Hong Kong, Hong Kong, Peoples R China
推荐引用方式 GB/T 7714	Gao, Bin,Liu, Xin,Chen, Xiaojun,et al. A NEW FIRST-ORDER ALGORITHMIC FRAMEWORK FOR OPTIMIZATION PROBLEMS WITH ORTHOGONALITY CONSTRAINTS[J]. SIAM JOURNAL ON OPTIMIZATION,2018,28(1):302-332.
APA	Gao, Bin,Liu, Xin,Chen, Xiaojun,&Yuan, Ya-Xiang.(2018).A NEW FIRST-ORDER ALGORITHMIC FRAMEWORK FOR OPTIMIZATION PROBLEMS WITH ORTHOGONALITY CONSTRAINTS.SIAM JOURNAL ON OPTIMIZATION,28(1),302-332.
MLA	Gao, Bin,et al."A NEW FIRST-ORDER ALGORITHMIC FRAMEWORK FOR OPTIMIZATION PROBLEMS WITH ORTHOGONALITY CONSTRAINTS".SIAM JOURNAL ON OPTIMIZATION 28.1(2018):302-332.