Publication Cover
Optimization
A Journal of Mathematical Programming and Operations Research
Volume 71, 2022 - Issue 15
378
Views
9
CrossRef citations to date
0
Altmetric
Research Article

Tseng's methods for inclusion problems on Hadamard manifolds

ORCID Icon, ORCID Icon, ORCID Icon & ORCID Icon
Pages 4367-4401 | Received 26 May 2020, Accepted 16 Apr 2021, Published online: 19 Jun 2021

References

  • Rockafellar RT. Monotone operators and the proximal point algorithm. SIAM J Control Optim. 1976;14(5):877–898. https://doi.org/10.1137/0314056.
  • Padcharoen A, Kitkuan D, Kumam W, et al. Tseng methods with inertial for solving inclusion problems and application to image deblurring and image recovery problems. Comput Math Methods. 2020 Mar. https://doi.org/10.1002/cmm4.1088..
  • Tseng P. A modified forward–backward splitting method for maximal monotone mappings. SIAM J Control Optim. 2000;38(2):431–446. https://doi.org/10.1137/S0363012998338806.
  • Aoyama K, Kimura Y, Takahashi W, et al. On a strongly nonexpansive sequence in Hilbert spaces. J Nonlinear Convex Anal. 2007;8(3):471–489.
  • Kitkuan D, Kumam P, Martínez-Moreno J, et al. Inertial viscosity forward–backward splitting algorithm for monotone inclusions and its application to image restoration problems. Int J Comput Math. 2020;97(1-2):482–497. https://doi.org/10.1080/00207160.2019.1649661.
  • Takahashi W, Wong NC, Yao JC. Two generalized strong convergence theorems of Halpern's type in Hilbert spaces and applications. Taiwanese J Math. 2012;16(3):1151–1172. https://doi.org/10.11650/twjm/1500406684.
  • Chang SS, Cho YJ, Lee BS, et al. Generalized set-valued variational inclusions in Banach spaces. J Math Anal Appl. 2000;246(2):409–422. https://doi.org/10.1006/jmaa.2000.6795.
  • Zhang C, Wang Y. Proximal algorithm for solving monotone variational inclusion. Optimization. 2018;67(8):1197–1209. https://doi.org/10.1080/02331934.2018.1455832.
  • Gibali A, Thong DV. Tseng type methods for solving inclusion problems and its applications. Calcolo. 2018;55(4):Paper No. 49, 22. https://doi.org/10.1007/s10092-018-0292-1.
  • Lions PL, Mercier B. Splitting algorithms for the sum of two nonlinear operators. SIAM J Numer Anal. 1979;16(6):964–979. https://doi.org/10.1137/0716071.
  • Passty GB. Ergodic convergence to a zero of the sum of monotone operators in Hilbert space. J Math Anal Appl. 1979;72(2):383–390. https://doi.org/10.1016/0022-247X(79)90234-8.
  • Ferreira OP, Oliveira PR. Proximal point algorithm on Riemannian manifolds. Optimization. 2002;51(2):257–270. https://doi.org/10.1080/02331930290019413.
  • Li C, López G, Martín-Márquez V. Monotone vector fields and the proximal point algorithm on Hadamard manifolds. J Lond Math Soc (2). 2009;79(3):663–683. https://doi.org/10.1112/jlms/jdn087.
  • Ansari QH, Babu F, Li XB. Variational inclusion problems in Hadamard manifolds. J Nonlinear Convex Anal. 2018;19(2):219–237.
  • Chen J, Liu S, Chang X. Modified Tseng's extragradient methods for variational inequality on Hadamard manifolds. Appl Anal. 2019;0(0):1–14. https://doi.org/10.1080/00036811.2019.1695783.
  • Wang JH, López G, Martín-Márquez V, et al. Monotone and accretive vector fields on Riemannian manifolds. J Optim Theory Appl. 2010;146(3):691–708. https://doi.org/10.1007/s10957-010-9688-z.
  • Li C, López G, Martín-Márquez V, et al. Resolvents of set-valued monotone vector fields in Hadamard manifolds. Set-Valued Var Anal. 2011;19(3):361–383. https://doi.org/10.1007/s11228-010-0169-1.
  • Bento GC, Ferreira OP, Oliveira PR. Proximal point method for a special class of nonconvex functions on Hadamard manifolds. Optimization. 2015;64(2):289–319. https://doi.org/10.1080/02331934.2012.745531.
  • Ansari QH, Babu F. Proximal point algorithm for inclusion problems in Hadamard manifolds with applications. Optim Lett. 2021;15(3):901--921. https://doi.org/10.1007/s11590-019-01483-0..
  • Németh SZ. Variational inequalities on Hadamard manifolds. Nonlinear Anal. 2003;52(5):1491–1498. https://doi.org/10.1016/S0362-546X(02)00266-3.
  • Chen J, Liu S. Extragradient-like method for pseudomontone equilibrium problems on Hadamard manifolds. J Inequal Appl. 2020;Paper No. 205, 15. https://doi.org/10.1186/s13660-020-02473-y.
  • Nguyen LV. Weak sharpness and finite termination for variational inequalities on Hadamard manifolds. Optimization. 2020 Feb;1–16. https://doi.org/10.1080/02331934.2020.1731807..
  • Al-Homidan S, Ansari QH, Babu F. Halpern- and Mann-type algorithms for fixed points and inclusion problems on Hadamard manifolds. Numerical Functional Analysis and Optimization. 2019;40(6):621–653. https://doi.org/10.1080/01630563.2018.1553887..
  • Ansari QH, Babu F, Yao JC. Regularization of proximal point algorithms in Hadamard manifolds. J Fixed Point Theory Appl. 2019;21(1):Paper No. 25, 23. https://doi.org/10.1007/s11784-019-0658-2.
  • Tang Gj, Huang Nj. Rate of convergence for proximal point algorithms on Hadamard manifolds. Oper Res Lett. 2014;42(6-7):383–387. https://doi.org/10.1016/j.orl.2014.06.009.
  • Ansari QH, Babu F. Existence and boundedness of solutions to inclusion problems for maximal monotone vector fields in Hadamard manifolds. Optim Lett. 2020;14(3):711–727. https://doi.org/10.1007/s11590-018-01381-x.
  • Sakai T. Riemannian geometry. Providence (RI): American Mathematical Society; 1996. (vol. 149 of Translations of mathematical monographs). Translated from the 1992 Japanese original by the author.
  • do Carmo MPa. Riemannian geometry. Mathematics: theory & applications. Boston (MA): Birkhäuser Boston, Inc.; 1992. Translated from the second Portuguese edition by Francis Flaherty. https://doi.org/10.1007/978-1-4757-2201-7.
  • Udrişte C. Convex functions and optimization methods on Riemannians manifolds. Dordrecht: Kluwer Academic Publishers Group; 1994. (vol. 297 of Mathematics and its applications). https://doi.org/10.1007/978-94-015-8390-9.
  • Bridson MR, Haefliger A. Metric spaces of non-positive curvature. Berlin: Springer-Verlag; 1999. (vol. 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental principles of mathematical sciences]). https://doi.org/10.1007/978-3-662-12494-9.
  • Németh SZ. Monotone vector fields. Publ Math Debrecen. 1999;54(3-4):437–449.
  • da Cruz Neto JX, Ferreira OP, Lucambio Pérez LR. Monotone point-to-set vector fields. Balkan J Geom Appl. 2000;5(1):69–79. Dedicated to Professor Constantin Udrişte.
  • Ferreira OP, Pérez LRL, Németh SZ. Singularities of monotone vector fields and an extragradient-type algorithm. J Global Optim. 2005;31(1):133–151. https://doi.org/10.1007/s10898-003-3780-y.
  • Tang Gj, Huang Nj. Korpelevich's method for variational inequality problems on Hadamard manifolds. J Global Optim. 2012;54(3):493–509. https://doi.org/10.1007/s10898-011-9773-3.
  • Absil PA, Mahony R, Sepulchre R. Optimization algorithms on matrix manifolds. Princeton (NJ): Princeton University Press; 2008. With a foreword by Paul Van Dooren. https://doi.org/10.1515/9781400830244.
  • Rothaus Oscar S. Domains of Positivity. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. 1960;24(1):189–235. http://dx.doi.org/10.1007/BF02942030.
  • Nesterov YE, Todd MJ. On the Riemannian geometry defined by self-concordant barriers and interior-point methods. Found Comput Math. 2002;2(4):333–361. https://doi.org/10.1007/s102080010032.
  • Lang S. Fundamentals of differential geometry. New York: Springer-Verlag; 1999. (vol. 191 of Graduate texts in mathematics). https://doi.org/10.1007/978-1-4612-0541-8.
  • Da Cruz Neto JX, Ferreira OP, Pérez LRL, et al. Convex- and monotone-transformable mathematical programming problems and a proximal-like point method. J Global Optim. 2006;35(1):53–69. https://doi.org/10.1007/s10898-005-6741-9.
  • Tang Gj, Zhou Lw, Huang Nj. The proximal point algorithm for pseudomonotone variational inequalities on Hadamard manifolds. Optim Lett. 2013;7(4):779–790. https://doi.org/10.1007/s11590-012-0459-7.

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.