非凸复合优化问题的黄金比率邻近交替线性化算法

doi:10.15960/j.cnki.issn.1007-6093.2025.02.006

运筹学学报（中英文） ›› 2025, Vol. 29 ›› Issue (2): 80-94.doi: 10.15960/j.cnki.issn.1007-6093.2025.02.006

非凸复合优化问题的黄金比率邻近交替线性化算法

曾康¹, 龙宪军¹^,*()

1. 重庆工商大学数学与统计学院, 重庆 400067

收稿日期:2024-07-21 出版日期:2025-06-15 发布日期:2025-06-12
通讯作者: 龙宪军 E-mail:xianjunlong@ctbu.edu.cn
基金资助:
重庆市自然科学基金(CSTB2024NSCQ-MSX1282);重庆市研究生导师团队建设项目(yds223010);重庆工商大学研究生创新型科研项目(yjscxx2025-269-238)

A golden ratio proximal alternating linearized algorithm for nonconvex composite optimization problems

Kang ZENG¹, Xianjun LONG¹^,*()

1. School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China

Received:2024-07-21 Online:2025-06-15 Published:2025-06-12
Contact: Xianjun LONG E-mail:xianjunlong@ctbu.edu.cn

摘要/Abstract

摘要：

本文考虑一类完全非凸的复合优化问题, 其目标函数由如下两部分组成: 关于全局变量不可分的连续可微非凸函数, 与两个关于独立变量的正常下半连续非凸函数。本文提出一种求解该问题的新型黄金比率邻近交替线性化极小化算法。在Kurdyka-Lojasiewicz (简记KL)性质假设下, 证明了由算法产生的迭代序列收敛到问题的稳定点。最后将新算法应用于求解稀疏信号恢复问题, 数值实验验证了新算法的有效性与优越性。

关键词: 非凸复合优化问题, 黄金比率邻近交替线性化算法, KL性质, 收敛性

Abstract:

In this paper, we consider a class of nonconvex composite optimization problems, whose objective function is the sum of a continuous differentiable bifunction of the entire variables, and two proper lower semi-continuous nonconvex function of their private variables. We propose a new golden ratio proximal alternating linearized algorithm to solve this problem. Under the assumption of Kurdyka-Lojasiewicz (in short: KL) property, we prove the iterative sequence generated by the algorithm converges to the critical point of the problem. Finally, numerical results on sparse signal recovery illustrate the efficiency and superiority of the proposed algorithm.

Key words: nonconvex composite optimization problem, golden ration proximal alternating linearized algorithm, KL property, convergence

中图分类号:

O224

曾康, 龙宪军. 非凸复合优化问题的黄金比率邻近交替线性化算法[J]. 运筹学学报（中英文）, 2025, 29(2): 80-94.

Kang ZENG, Xianjun LONG. A golden ratio proximal alternating linearized algorithm for nonconvex composite optimization problems[J]. Operations Research Transactions, 2025, 29(2): 80-94.

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算法	iter=500		iter=1 000		iter=1 500
算法	Time/s	res	Time/s	res	Time/s	res
算法1	18.259 4	0.245 0	36.343 8	0.225 0	54.802 5	0.222 8
GiPALM	18.406 3	0.982 6	36.703 1	0.983 9	54.921 9	0.984 4
PALM	18.546 9	1.045 5	37.312 5	0.983 8	56.283 1	0.993 6
NiPALM	18.781 3	1.731 7	36.656 3	1.701 3	56.375 0	1.701 3

算法	err=10^-3		err=10^-4		err=10^-5
算法	Time/s	res	Time/s	res	Time/s	res
算法1	18.259 4	0.245 0	18.012 0	0.245 0	18.000 0	0.236 5
GiPALM	18.406 3	0.982 6	18.156 3	0.982 6	18.493 8	0.982 6
PALM	18.546 9	1.045 5	18.593 8	1.045 5	18.546 9	1.045 5
NiPALM	18.781 3	1.731 7	19.140 6	1.731 7	19.171 9	1.632 0

算法	err=10^-3		err=10^-4		err=10^-5		err=10^-6
算法	Time/s	Iter	Time/s	Iter	Time/s	Iter	Time/s	Iter
算法1	2.125 0	56	3.468 8	92	5.281 3	130	6.328 1	170
PG	13.812 5	386	18.125 0	347	27.609 4	832	36.390 6	1 004
FPG	8.062 5	262	10.609 4	508	11.265 6	370	18.734 4	542

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非凸复合优化问题的黄金比率邻近交替线性化算法

A golden ratio proximal alternating linearized algorithm for nonconvex composite optimization problems

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