一种新的求解单调算子零点问题的预测-校正算法框架

doi:10.15960/j.cnki.issn.1007-6093.2026.03.010

Abstract

Abstract: This paper proposes a novel predictor-corrector algorithm framework for monotone inclusion problems. Many classes of problems, such as multi-block separable convex optimization problems and saddle point problems, can be reformulated as monotone inclusions, and algorithms can be designed via the proposed framework. The framework not only encompasses the relaxed forward-backward splitting method and the relaxed Davis-Yin splitting method, but also reproduces Algorithm 3.1 and Algorithm 5.1 in Chang et al (2025) by appropriately selecting the predictor steps and correction matrices, thereby simplifying their convergence analysis.

Key words: monotone inclusion problems, predictor-corrector algorithm framework, splitting methods

CLC Number:

O221.2

LUO Yueying, LIANG Tianzhe, CAI Xingju. A new predictor-corrector algorithm framework for solving monotone inclusion problems[J]. Operations Research Transactions, 2026, 30(3): 170-182.

Add to citation manager EndNote|Reference Manager|ProCite|BibTeX|RefWorks

URL: https://www.ort.shu.edu.cn/EN/10.15960/j.cnki.issn.1007-6093.2026.03.010

https://www.ort.shu.edu.cn/EN/Y2026/V30/I3/170

[1] Zhu M, Chan T. An efficient primal-dual hybrid gradient algorithm for total variation image restoration [R]. CAM Report 08-34. Los Angeles: University of California, 2008.
[2] Esser E, Zhang X, Chan T F. A general framework for a class of first-order primal-dual algorithms for convex optimization in imaging science [J]. SIAM Journal on Imaging Sciences, 2010, 3(4): 1015-1046.
[3] Chambolle A, Pock T. A first-order primal-dual algorithm for convex problems with applications to imaging [J]. Journal of Mathematical Imaging and Vision, 2011, 40(1): 120-145.
[4] Cai X, Han D, Xu L. An improved first-order primal-dual algorithm with a new correction step [J]. Journal of Global Optimization, 2013, 57(4): 1419-1428.
[5] He B, Ma F, Xu S, et al. A generalized primal-dual algorithm with improved convergence condition for saddle point problems [J]. SIAM Journal on Imaging Sciences, 2022, 15(3): 1157-1183.
[6] Martinet B. Régularisation d’inéquations variationnelles par approximations successives [J]. Revue fra?caise d’informatique et de recherche opérationnelle. Série rouge, 1970, 4(R3): 154- 158.
[7] Rockafellar R T. Monotone operators and the proximal point algorithm [J]. SIAM Journal on Control and Optimization, 1976, 14(5): 877-898.
[8] Lions P L, Mercier B. Splitting algorithms for the sum of two nonlinear operators [J]. SIAM Journal on Numerical Analysis, 1979, 16(6): 964-979.
[9] Jia Z, Cai X. A relaxation of the parameter in the forward-backward splitting method [J]. Pacific Journal of Optimization, 2017, 13(4): 665-681.
[10] Chang X, Yang J, Bai J, et al. A primal-dual splitting algorithm with convex combination and larger step sizes for composite monotone inclusion problems [J]. SIAM Journal on Imaging Sciences, 2025, 18(4): 2094-2126.
[11] He B S. PPA-like contraction methods for convex optimization: a framework using variational inequality approach [J]. Journal of the Operations Research Society of China, 2015, 3(4): 391- 420.
[12] Bauschke H H, Combettes P L. Convex analysis and monotone operator theory in Hilbert spaces [M]. Cham: Springer International Publishing, 2017.
[13] Douglas J, Rachford H H. On the numerical solution of heat conduction problems in two and three space variables [J]. Transactions of the American Mathematical Society, 1956, 82(2): 421-439.
[14] Eckstein J, Bertsekas D P. On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators [J]. Mathematical Programming, 1992, 55(1): 293- 318.
[15] Bredies K, Sun H. Preconditioned Douglas-Rachford splitting methods for convex-concave saddle-point problems [J]. SIAM Journal on Numerical Analysis, 2015, 53(1): 421-444.
[16] Davis D, Yin W. A three-operator splitting scheme and its optimization applications [J]. SetValued and Variational Analysis, 2017, 25(4): 829-858.
[17] He B, Yuan X. Convergence analysis of primal-dual algorithms for a saddle-point problem: from contraction perspective [J]. SIAM Journal on Imaging Sciences, 2012, 5(1): 119-149.
[18] Arrow K J, Hurwicz L, Uzawa H. Studies in linear and non-linear programming [M]. Stanford: Stanford University Press, 1958.
[19] He B, You Y, Yuan X. On the convergence of primal-dual hybrid gradient algorithm [J]. SIAM Journal on Imaging Sciences, 2014, 7(4): 2526-2537.
[20] Korpelevich G M. An extragradient method for finding saddle points and for other problems [J]. Ekonomika i Matematicheskie Metody, 1976, 12: 747-756.
[21] Censor Y, Gibali A, Reich S. Extensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space [J]. Optimization, 2012, 61(9): 1119-1132.
[22] Popov L D. A modification of the Arrow-Hurwicz method for search of saddle points [J]. Mathematical Notes of the Academy of Sciences of the USSR, 1980, 28(5): 845-848.
[23] Solodov M V, Svaiter B F. A new projection method for variational inequality problems [J]. SIAM Journal on Control and Optimization, 1999, 37(3): 765-776.
[24] Wang Y, Xiu N, Wang C. Unified framework of extragradient-type methods for pseudomonotone variational inequalities [J]. Journal of Optimization Theory and Applications, 2001, 111(3): 641-656.
[25] Han D. A modified alternating direction method for variational inequality problems [J]. Applied Mathematics & Optimization, 2002, 45(1): 63-74.
[26] Gao X, Cai X, Wang X. Self-adaptive inertial projection and contraction algorithm for monotone variational inequality [J]. Asia-Pacific Journal of Operational Research, 2022, 39(2): 2150021.
[27] Tseng P. A modified forward-backward splitting method for maximal monotone mappings [J]. SIAM Journal on Control and Optimization, 2000, 38(2): 431-446.
[28] Malitsky Y, Tam M. A forward-backward splitting method for monotone inclusions without cocoercivity [J]. SIAM Journal on Optimization, 2020, 30(2): 1451-1472.
[29] Ryu E K, V u B C. Finding the forward-Douglas-Rachford-forward method [J]. Journal of Optimization Theory and Applications, 2020, 184(3): 858-876.
[30] Aragón-Artacho F J, Torregrosa-Belén D. A direct proof of convergence of Davis{Yin splitting algorithm allowing larger stepsizes [J]. Set-Valued and Variational Analysis, 2022, 30(3): 1011- 1029.

A new predictor-corrector algorithm framework for solving monotone inclusion problems

PDF

Knowledge

Abstract

Cite this article

share this article

Related Articles 15

Recommended Articles

Metrics

Comments

[1]	GUO Yanjing, GAO Pan, WU Zhongming. Two-stage drone facility location and scheduling based on robust satisficing [J]. Operations Research Transactions, 2026, 30(3): 204-218.
[2]	WANG Xiangfeng, ZENG Shangzhi, ZHANG Jin, ZHOU Jinchuan. Proximal-based methods can guarantee blunt local minimizer for nonconvex nonsmooth optimization problem [J]. Operations Research Transactions, 2026, 30(2): 1-23.
[3]	MEN Yanchao, LI Xudong. Fast algorithm for OWL1 norm constrained regression model [J]. Operations Research Transactions, 2026, 30(2): 24-44.
[4]	ZENG Jing, XIANG Yao, ZHANG Wenyan. The well-posedness of traffic equilibrium problems [J]. Operations Research Transactions, 2026, 30(2): 58-68.
[5]	WANG Feng, HANG Bo, HUANG Jinzhou, XU Degang, ZHANG Zeyu, LIU Jiamou. Research of touring route planning based on spatio-temporal awareness [J]. Operations Research Transactions, 2026, 30(2): 69-78.
[6]	WU Xiaoyu, SHAO Hu, LIU Pengjie, ZHOU Jincheng. Two RMIL-type conjugate gradient methods with sufficient descent property and applications in image restoration [J]. Operations Research Transactions, 2026, 30(2): 79-92.
[7]	CUI Hengxin, JIANG Fan. Inexact proximal point algorithms and projection methods for monotone variational inequalities [J]. Operations Research Transactions, 2026, 30(2): 194-208.
[8]	ZHANG Xuefeng, PENG Xiao, CHEN Liangyu, YANG Zhengfeng, ZENG Zhenbing. A review of intelligent branch and bound algorithms for mixed integer linear programming problems [J]. Operations Research Transactions, 2026, 30(2): 237-270.
[9]	WEI Jiazhen, BIAN Wei. A survey on research advances in consensus-based optimization algorithm [J]. Operations Research Transactions, 2026, 30(1): 1-23.
[10]	ZHOU Yunxu, YAO Fanjun, GAO Hongwei. Blockchain empowering the digital transformation: Study on the coordination strategies of supply chain [J]. Operations Research Transactions, 2026, 30(1): 61-74.
[11]	ZHAO Hongxin, KONG Lingchen. Research on diversification portfolio optimization model and method [J]. Operations Research Transactions, 2026, 30(1): 75-92.
[12]	SUN Ke, WANG Jinting, WANG Zhongbin. The economics of waiting-area entertainment [J]. Operations Research Transactions, 2026, 30(1): 93-107.
[13]	XU Ke, JI Lanping, GONG Hua, LIU Peng, SUN Wenjuan. A coordinated multi-agent production and transportation scheduling on parallel machines based on auction algorithm [J]. Operations Research Transactions, 2026, 30(1): 121-136.
[14]	WANG Ziqi, WANG Junlin, XU Zi. A variance reduced gradient descent ascent algorithm for a class of nonconvex-nonconcave minimax problems [J]. Operations Research Transactions, 2026, 30(1): 197-206.
[15]	YANG Jinji, SHEN Chungen, YU Zhensheng. Convergence analysis of an adaptive proximal gradient-subgradient algorithm for square-root-loss regression problems [J]. Operations Research Transactions, 2026, 30(1): 217-234.