Publication Cover
Optimization
A Journal of Mathematical Programming and Operations Research
Volume 73, 2024 - Issue 7
413
Views
6
CrossRef citations to date
0
Altmetric
Research Article

Modified accelerated Bregman projection methods for solving quasi-monotone variational inequalities

, ORCID Icon, ORCID Icon &
Pages 2053-2087 | Received 22 Oct 2022, Accepted 28 Feb 2023, Published online: 15 Mar 2023

References

  • Cottle RW, Yao JC. Pseudo-monotone complementarity problems in Hilbert space. J Optim Theory Appl. 1992;75:281–295.
  • Ye M, He Y. A double projection method for solving variational inequalities without monotonicity. Comput Optim Appl. 2015;60:141–150.
  • Fichera G. Sul problema elastostatico di Signorini con ambigue condizioni al contorno. Atti Accad Naz Lincci VIII Scr Rend Cl Sci Fis Mat Nat. 1963;34:138–142.
  • Fichera G. Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno. Atti Accad Naz Lincci Mem Cl Sci Fis Mat Nat. 1964;7:91–140.
  • Stampacchia G. Formes bilinearieres coercitivities sur les ensembles convexes. CR Acad Sci Paris. 1964;258:4413–4416.
  • Baiocchi C, Capelo A. Variational and quasivariational inequalities: applications to free boundary problems. New York (NY): Wiley; 1984.
  • Bensoussan A, Lions JL. Applications of variational inequalities to stochastic control. Amsterdam (AMS): North-Holland; 1982.
  • Facchinei F, Pang JS. Finite-dimensional variational inequalities and complementarity problem. New York (NY): Springer; 2003.
  • Grannessi F, Mangeri A. Variational inequalities and network equilibrium problems. New York (NY): Plenum Press; 1995.
  • Khobotov EN. Modification of the extragradient method for solving variational inequalities and certain optimization problems. USSR Comput Math & Math Phys. 1989;27:120–127.
  • Kinderlehrer D, Stampacchia G. An introduction to variational inequalities and their applications. New York (NY): Academic Press Inc; 1980.
  • Nagurney A. Network economics: a variational inequality approach. Dordrecht: Kluwer Academic Publishers; 1989.
  • He BS, Liao LZ. Improvements of some projection methods for monotone nonlinear variational inequalities. J Optim Theory Appl. 2002;112:111–128.
  • Korpelevich GM. The extragradient method for finding saddle points and other problems. Ekon Mate Metody. 1976;12:747–756.
  • Censor Y, Gibali A, Reich S. The subgradient extragradient method for solving variational inequalities in Hilbert space. J Optim Theory Appl. 2011;148:318–335.
  • Tseng P. A modified forward-backward splitting method for maximal monotone mapping. SIAM J Control Optim. 2000;38:431–446.
  • Shehu Y, Dong QL, Jiang D. Single projection method for pseudo-monotone variational inequality in Hilbert spaces. Optimization. 2019;68(1):385–409.
  • Thong DV, Vinh NT, Cho YJ. Accelerated subgradient extragradient methods for variational inequality problems. J Sci Comput. 2019;80:1438–1462.
  • Vuong PT. On the weak convergence of the extragradient method for solving pseudo-monotone variational inequalities. J Optim Theory Appl. 2018;176:399–409.
  • Zhao TY, Wang DQ, Ceng LC, et al. Quasi-inertial Tseng's extragradient algorithms for pseudomonotone variational inequalities and fixed point problems of quasi-nonexpansive operators. Numer Funct Anal Optim. 2020;42:69–90.
  • Hieu DV, Anh PK, Muu LD. Modified extragradient-like algorithms with new stepsizes for variational inequalities. Comput Optim Appl. 2019;73:913–932.
  • Alakoya TO, Mewomo OT, Shehu Y. Strong convergence results for quasimonotone variational inequalities. Math Methods Oper Res. 2021;95:249–279.
  • Hieu DV, Cholamjiak P. Modified extragradient method with Bregman distance for variational inequalities. Appl Anal. 2022;101(2):655–670.
  • Hieu DV, Reich S. Two Bregman projection methods for solving variational inequalities. Optimization. 2022;71(7):1777–1802.
  • Liu H, Yang J. Weak convergence of iterative methods for solving quasimonotone variational inequalities. Comput Optim Appl. 2020;77:491–508.
  • Salahuddin. The extragradient method for quasi-monotone variational inequalities. Optimization. 2020;71(9):2519–2528.
  • Polyak BT. Some methods of speeding up the convergence of iterates methods. USSR Comput Math Phys. 1964;4(5):1–17.
  • Attouch H, Maing e´ PE. Asymptotic behavior of second-order dissipative equations evolution combining potential with non-potential effects. ESAIM Contr Optim Calc. 2011;17:836–857.
  • Boţ RI, Csetnek ER. Second order forward-backward dynamical systems for monotone inclusion problems. SIAM J Control Optim. 2016;54:1423–1443.
  • Attouch H, Cabot A. Convergence rate of a relaxed inertial proximal algorithm for convex minimization. Optimization. 2020;69(6):1281–1312.
  • Ceng LC, Petrusel A, Qin X, et al. A modified inertial subgradient extragradient method for solving pseudomonotone variational inequalities and common fixed point problems. Fixed Point Theory. 2020;21:93–108.
  • Ceng LC, Petrusel A, Qin X, et al. Two inertial subgradient extragradient algorithms for variational inequalities with fixed-point constraints. Optimization. 2021;70:1337–1358.
  • Ceng LC, Shang M. Hybrid inertial subgradient extragradient methods for variational inequalities and fixed point problems involving asymptotically nonexpansive mappings. Optimization. 2021;70:715–740.
  • Ogwo GN, Izuchukwu C, Shehu Y, et al. Convergence of relaxed inertial subgradient extragradient methods for quasimonotone variational inequality problems. J Sci Comput. 2022;90:10.
  • Reich S, Tuyen TM, Sunthrayuth S, et al. Two new inertial algorithms for solving variational inequalities in reflexive Banach spaces. Numer Funct Anal Optim. 2021;42(16):1954–1984.
  • Tang Y, Sunthrayuth P. An iterative algorithm with inertial technique for solving the split common null point problem in Banach spaces. Asian-Eur J Math. 2022;15(6):2250120.
  • Tuyen TM, Sunthrayuth P, Trang NM. An inertial self-adaptive algorithm for the generalized split common null point problem in Hilbert spaces. Rend Circ Mat Palermo. 2022;71(2):537–557.
  • Wang ZB, Chen X, Yi J, et al. Inertial projection and contraction algorithms with larger step sizes for solving quasimonotone variational inequalities. J Glob Optim. 2022;82:499–522.
  • Chbani Z, Riahi H. Weak and strong convergence of an inertial proximal method for solving Ky Fan minimax inequalities. Optim Lett. 2013;7:185–206.
  • Bregman LM. The relaxation method for finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput Math Math Phys. 1967;7(3):200–217.
  • Amid E, Warmuth MK, Anil R, et al. Robust bi-tempered logistic loss based on Bregman divergences. In: Conference on Neural Information Processing Systems. Vancouver; 2019. p. 14987–14996.
  • Gray RM, Buzo A, Gray AH, et al. Distortion measures for speech processing. IEEE Trans Acoust Speech Signal Process. 1980;28:367–376.
  • Cesa-Bianchi N, Lugosi G. Prediction, learning, and games. New York: Cambridge University Press; 2006.
  • Banerjee A, Merugu S, Dhillon I, et al. Clustering with Bregman divergences. J Mach Learn Res. 2005;6:1705–1749.
  • Fischer A. Quantization and clustering with Bregman divergences. J Multivar Anal. 2010;101:2207–2221.
  • Chen JZ, Hu HY, Ceng LC. Strong convergence of hybrid Bregman projection algorithm for split feasibility and fixed point problems in Banach spaces. J Nonlinear Sci Appl. 2017;10:192–204.
  • Gibali A. A new Bregman projection method for solving variational inequalities in Hilbert spaces. Pure Appl Funct Anal. 2018;3:403–415.
  • Jolaoso LO, Oyewole OK, Aremu KO. A Bregman subgradient extragradient method with self-adaptive technique for solving variational inequalities in reflexive Banach spaces. Optimization. 2022;71:3835–3860. DOI: 10.1080/02331934.2021.1925669 .
  • Sunthrayuth P, Jolaoso LO, Cholamjiak P. New Bregman projection methods for solving pseudo-monotone variational inequality problem. J Appl Math Comput. 2022;68:1565–1589.
  • Alber Y, Butnariu D. Convergence of Bregman projection methods for solving consistent convex feasibility problems in reflexive Banach spaces. J Optim Theory Appl. 1997;92:33–61.
  • Pathak HK. An introduction to nonlinear analysis and fixed point theory. Singapore: Springer; 1980.
  • Reich S, Sabach S. A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces. J Nonlinear Convex Anal. 2009;1:471–485.
  • Z a˘linescu C. Convex analysis in general vector spaces. Singapore: World Scientific; 2002.
  • Beck A. First-order methods in optimization. Philadelphia (PA): Society for Industrial and Applied Mathematics; 2017.
  • Bauschke HH, Borwein JM, Combettes PL. Essential smoothness, essential strict convexity, and legendre functions in Banach spaces. Commun Contemp Math. 2001;3:615–647.
  • Beck A, Teboulle M. Mirror descent and nonlinear projected subgradient methods for convex optimization. Oper Res Lett. 2003;31(3):167–175.
  • Bauschke HH, Bolte J, Teboulle M. A descent lemma beyond Lipschitz gradient continuity: first-order methods revisited and applications. Math Oper Res. 2016;42(2):330–348.
  • Bauschke HH, Borwein JM. Legendre functions and the method of random Bregman projections. J Convex Anal. 1997;4:27–67.
  • Bolte J, Sabach S, Teboulle M, et al. First order methods beyond convexity and Lipschitz gradient continuity with applications to quadratic inverse problems. SIAM J Control Optim. 2018;28(3):2131–2151.
  • Mukkamala MC, Ochs P. Beyond alternating updates for matrix factorization with inertial Bregman proximal gradient algorithms. Adv Neural Inf Process Syst. 2019;4266–4276.
  • Teboulle M, Vaisbourd Y. Novel proximal gradient methods for nonnegative matrix factorization with sparsity constraints. SIAM J Imaging Sci. 2020;13(1):381–421.
  • Butnairu D, Resmerita E. Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces. Abstr Appl Anal. 2006;2006:1–39. Article ID 84919.
  • Martín-M a´rquez V, Reich S, Sabach S. Iterative methods for approximating fixed points of Bregman nonexpansive operators. Discrete Contin Dyn Syst-S. 2013;6(4):1043–1063.
  • Huang YY, Jeng JC, Kuo TY, et al. Fixed point and weak convergence theorems for point-dependent λ-hybrid mappings in Banach spaces. Fixed Point Theory Appl. 2011;2011:105.
  • Tan KK, Xu HK. Approximating fixed points of nonexpensive mappings by the Ishikawa iteration process. J Math Anal Appl. 1993;178(2):301–308.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.