A Golden Ratio Primal–Dual Algorithm for Structured Convex Optimization

doi:10.1007/s10915-021-01452-9

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	A Golden Ratio Primal–Dual Algorithm for Structured Convex Optimization
	Chang, Xiaokai 2; Yang, Junfeng 1
刊名	Journal of Scientific Computing
	2021-05-01
卷号	87 期号:2
关键词	Convex optimization Game theory Mathematical transformations Matrix algebra Regression analysis Convex combinations Ergodic convergence Matrix vector multiplication Nonnegative least squares Number of iterations Objective functions Regularized least squares Structured convex optimizations
ISSN号	0885-7474
DOI	10.1007/s10915-021-01452-9
英文摘要	We design, analyze and test a golden ratio primal–dual algorithm (GRPDA) for solving structured convex optimization problem, where the objective function is the sum of two closed proper convex functions, one of which involves a composition with a linear transform. GRPDA preserves all the favorable features of the classical primal–dual algorithm (PDA), i.e., the primal and the dual variables are updated in a Gauss–Seidel manner, and the per iteration cost is dominated by the evaluation of the proximal point mappings of the two component functions and two matrix-vector multiplications. Compared with the classical PDA, which takes an extrapolation step, the novelty of GRPDA is that it is constructed based on a convex combination of essentially the whole iteration trajectory. We show that GRPDA converges within a broader range of parameters than the classical PDA, provided that the reciprocal of the convex combination parameter is bounded above by the golden ratio, which explains the name of the algorithm. An O(1 / N) ergodic convergence rate result is also established based on the primal–dual gap function, where N denotes the number of iterations. When either the primal or the dual problem is strongly convex, an accelerated GRPDA is constructed to improve the ergodic convergence rate from O(1 / N) to O(1 / N2). Moreover, we show for regularized least-squares and linear equality constrained problems that the reciprocal of the convex combination parameter can be extended from the golden ratio to 2 and meanwhile a relaxation step can be taken. Our preliminary numerical results on LASSO, nonnegative least-squares and minimax matrix game problems, with comparisons to some state-of-the-art relative algorithms, demonstrate the efficiency of the proposed algorithms. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
WOS研究方向	Mathematics
语种	英语
出版者	Springer
WOS记录号	WOS:000631706100001
内容类型	期刊论文
源URL	[http://ir.lut.edu.cn/handle/2XXMBERH/148384]
专题	理学院
作者单位	1.Department of Mathematics, Nanjing University, Nanjing, China 2.School of Science, Lanzhou University of Technology, Lanzhou; Gansu, China;
推荐引用方式 GB/T 7714	Chang, Xiaokai,Yang, Junfeng. A Golden Ratio Primal–Dual Algorithm for Structured Convex Optimization[J]. Journal of Scientific Computing,2021,87(2).
APA	Chang, Xiaokai,&Yang, Junfeng.(2021).A Golden Ratio Primal–Dual Algorithm for Structured Convex Optimization.Journal of Scientific Computing,87(2).
MLA	Chang, Xiaokai,et al."A Golden Ratio Primal–Dual Algorithm for Structured Convex Optimization".Journal of Scientific Computing 87.2(2021).