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Optimization
A Journal of Mathematical Programming and Operations Research
Volume 71, 2022 - Issue 13
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Research Article

Convergence analysis of new inertial method for the split common null point problem

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Pages 3767-3795 | Received 05 Aug 2020, Accepted 01 Apr 2021, Published online: 14 Apr 2021

References

  • Censor Y, Gibali A, Reich S, Algorithms for the split variational inequality problem. Numer Algorithms. 2012;59:301–323.
  • Moudafi A. Split monotone variational inclusions. J Optim Theory Appl. 2011;150:275–283.
  • Censor Y, Elfving T. A multiprojection algorithm using Bregman projections in product space. Numer Algorithms. 1994;8:221–239.
  • Byrne C. Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 2002;18(2):441–453.
  • Moudafi A, Thakur BS. Solving proximal split Feasibility problem without prior knowledge of matrix norms. Optim Lett. 2014;8(7):2099–2110.
  • Gibali A, Mai DT, Nguyen TV. A new relaxed CQ algorithm for solving split feasibility problems in Hilbert spaces and its applications. J Ind Manag Optim. 2018;2018:1–25.
  • Cegielski A, Reich S, Zalas R. Weak, strong and linear convergence of the CQ-method via the regularity of Landweber operators. Optimization. 2020;69(3):605–636.
  • Jeong T, Jung YM, Yun S. Iterative reweighted algorithm for non-convex Poissonian image restoration model. J Korean Math Soc. 2018;55(3):719–734.
  • Reich S, Truong MT. Two projection methods for solving the multiple-set split common null point problem in Hilbert spaces. Optimization. 2020;69(9):1913–1934.
  • Reich S, Truong MT. Two projection algorithms for solving the split common fixed point problem. J Optim Theory Appl. 2020;186(1):148–168.
  • Tang Y, Gibali A. New self-adaptive step size algorithms for solving split variational inclusion problems and its applications. Numer Algorithms. 2020;83(1):305–331.
  • Zhao J, Hou D. A self-adaptive iterative algorithm for the split common fixed point problems. Numer Algorithms. 2019;82(3):1047–1063.
  • Byrne C, Censor Y, Gibali A, et al. Weak and strong convergence of algorithms for the split common null point problem. J Nonlinear Convex Anal. 2012;13:759–775.
  • Takahashi W. The split common null point problem for generalized resolvents in two Banach spaces. Numer Algorithms. 2017;75:1065–1078.
  • Takahashi S, Takahashi W. The split common null point problem and the shrinking projection method in Banach spaces. Optimization. 2016;65(2):281–287.
  • Takahashi W, Yao J-C. Strong convergence theorems by hybrid methods for the split common null point problem in Banach spaces. Fixed Point Theory Appl. 2015;2015: Article ID 87.
  • Polyak BT. Some methods of speeding up the convergence of iteration methods. USSR Comput Math Math Phys. 1964;4(5):1–17.
  • Alvarez F. On the minimizing property of a second order dissipative system in Hilbert spaces. SIAM J Control Optim. 2000;38(4):1102–1119.
  • Alvarez F. Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space. SIAM J Optim. 2004;14:773–782.
  • Alvarez F, Attouch H. An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal. 2001;9:3–11.
  • Bot RI, Csetnek ER, Laszlo SC. An inertial forward-backward algorithm for the minimization of the sum of two nonconvex functions. EURO J Comput Optim. 2016;4:3–25.
  • Bot RI, Csetnek ER. An inertial Tseng's type proximal algorithm for nonsmooth and nonconvex optimization problems. J Optim Theory Appl. 2015;17:600–616.
  • Bot RI, Csetnek ER. An inertial forward-backward-forward primal-dual splitting algorithm for solving monotone inclusion problems. Numer Algorithms. 2016;71: 519–540.
  • Bot RI, Csetnek ER, Hendrich C. Inertial Douglas–Rachford splitting for monotone inclusion problems. Appl Math Comput. 2015;256:472–487.
  • Bot RI, Csetnek ER. An inertial alternating direction method of multipliers. Minimax Theory Appl. 2016;1:29–49.
  • Chen C, Ma S, Yang J, A general inertial proximal point algorithm for mixed variational inequality problem. SIAM J Optim. 2015;25:2120–2142.
  • Maingé PE. Convergence theorems for inertial KM-type algorithms. J Comput Appl Math. 2008;219(1):223–236.
  • Maingé PE, Gobinddass ML. Convergence of one step projected gradient methods for variational inequalities. J Optim Theory Appl. 2016;171:146–168.
  • Maingé PE. Numerical approach to monotone variational inequalities by a one-step projected reflected gradient method with line-search procedure. Comput Math Appl. 2016;3:720–728.
  • Moudafi A, Elisabeth E. An approximate inertial proximal method using enlargement of a maximal monotone operator. Int J Pure Appl Math. 2003;5:283–299.
  • Moudafi A, Oliny M. Convergence of a splitting inertial proximal method for monotone operators. J Comput Appl Math. 2003;155:447–454.
  • Long LV. New algorithms for the split variational inclusion problems and application to split feasibility problems. Optimization. 2019;68(12):233–2367.
  • Chuang CS. Hybrid inertial proximal algorithm for the split variational inclusion problem in Hilbert spaces with applications. Optimization. 2017;66(5):777–792.
  • Majee P, Nahak C. On inertial proximal algorithm for split variational inclusion problems. Optimization. 2018;67(10):1701–1716.
  • Beck A, Teboulle M. A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J Imaging Sci. 2009;2:183–202.
  • Lorenz DA, Pock T. An inertial forward-backward algorithm for monotone inclusions. J Math Imaging Vision. 2015;51:311–325.
  • Malitsky Y, Pock T. A first-order primal-dual algorithm with line search. SIAM J Optim. 2018;28:411–432.
  • Mu Z, Peng Y. A note on the inertial proximal point method. Stat Optim Inf Comput. 2015;3(3):241–248.
  • Iutzeler F, Hendrickx JM. A generic online acceleration scheme for optimization algorithms via relaxation and inertia. Optim Methods Softw. 2019;34(2):383–405.
  • Iutzeler F, Malick J. On the proximal gradient algorithm with alternated inertia. J Optim Theory Appl. 2018;176(3):688–710.
  • Shehu Y, Gibali A. New inertial relaxed method for solving split feasibilities (in press). Optim Lett. doi:10.1007/s11590-020-01603-1.
  • Shehu Y, Iyiola OS. Projection methods with alternating inertial steps for variational inequalities: weak and linear convergence. Appl Numer Math. 2020;157:315–337.
  • Opial Z. Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull Am Math Soc. 1967;73:591–598.
  • Tang Y. New algorithms for the split common null point problem. Optimization. 2020; DOI:10.1080/02331934.2020.1782908.
  • Burke JV, Ferris MC. A Gauss-Newton method for convex composite optimization. Math Program. 1995;71:179–94.
  • Hu Y, Li C, Yang X. On convergence rates of linearized proximal algorithms for convex composite optimization with applications. SIAM J Optim. 2016;26:1207–1235.
  • Wang J, Hu Y, Li C, et al. Linear convergence of CQ algorithms and applications in gene regulatory network inference. Inverse Probl. 2017;33:055017.
  • Contreras J, Klusch M, Krawczyk JB. Numerical solution to Nash-Cournot equilibria in coupled constraint electricity markets. IEEE Trans Power Syst. 2004;19:195–206.
  • Wei JY, Yves S. Spatial oligopolistic electricity models with Cournot generators and regulated transmission prices. Oper Res. 1999;47(2):102–112.
  • Yen LH, Huyen NTT, Muu LD. A subgradient algorithm for a class of nonlinear split feasibility problems: application to jointly constrained Nash equilibrium models. J Global Optim. 2019;73:849–868.
  • Gorbachuk VM. Cournot-Nash and Bertrand-Nash equilibria for a heterogeneous duopoly of differentiated products. Cybern Syst Anal. 2010;46(1):25–33 (translated from KIBERnet System Anal. 2010;46(1):29–37).
  • Iusem AN. On some properties of paramonotone operator. Convex Anal. 1981;5:269–278.
  • Dong QL, Li X, Rassias ThM. Two projection algorithms for a class of split feasibility problems with jointly constrained Nash equilibrium models. Optimization. 2021;70:871–897. doi:10.1080/02331934.2020.1753741
  • Wang F, Xu HK. Cyclic algorithms for split feasibility problems in Hilbert spaces. Nonlinear Anal. 2011;74:4105–4111.

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