Publication Cover
Optimization
A Journal of Mathematical Programming and Operations Research
Volume 66, 2017 - Issue 7
149
Views
4
CrossRef citations to date
0
Altmetric
Original Articles

New results on the proximal point algorithm in non-positive curvature metric spaces

, &
Pages 1191-1199 | Received 03 May 2016, Accepted 26 Apr 2017, Published online: 16 May 2017

References

  • Martinet B. Régularisation d’Inéquations Variationnelles par Approximations Successives. Rev Fr Inform Rech O. 1970;3:154–158.
  • Rockafellar T. Monotone operators and the proximal point algorithm. SIAM J Control Optim. 1976;14:877–898.
  • Brezis H, Lions PL. Produits infinis de résolvantes. Israel J Math. 1978;29:329–345.
  • Güler O. On the convergence of the proximal point algorithm for convex minimization. SIAM J Control Optim. 1991;29:403–419.
  • Xu HK. Iterative algorithms for nonlinear operators. J London Math Soc. 2002;66:240–256.
  • Kamimura S, Takahashi W. Approximating solutions of maximal monotone operators in Hilbert Spaces. J Approx Theory. 2000;106:226–240.
  • Ahmadi P, Khatibzadeh H. On the convergence of inexact proximal point algorithm on Hadamard manifolds. Taiwanese J Math. 2014;18:419–433.
  • Bento GC, CruzNeto JX, Oliveira PR. A new approach to the proximal point method: Convergence on general Riemannian manifolds. J Optim Theory Appl. 2016;168:743–755.
  • Ferreira OP, Oliveira PR. Proximal point algorithm on Riemannian manifolds. Optimization. 2002;51:257–270.
  • Li C, Lopez G, Martin-Marquez V. Monotone vector fields and the proximal point algorithm on Hadamard manifolds. J London Math Soc. 2009;79:663–683.
  • Bacak M. The proximal point algorithm in metric spaces. Israel J Math. 2013;194:689–701.
  • Khatibzadeh H, Ranjbar S. Monotone operators and the proximal point algorithm in complete CAT(0) metric spaces. J Aust Math Soc. 2016. DOI:10.1017/s1446788716000446
  • Jost J. Convex functionals and generalized harmonic maps into spaces of non positive curvature. Comment Math Helvetici. 1995;70:659–673.
  • Jost J. Nonpositive curvature: geometric and analytic aspects. Lectures in mathematics. ETH Zürich. Basel: Birkhäuser; 1997.
  • Bacak M. Convex analysis and optimization in Hadamard spaces. Vol. 22, De Gruyter series in nonlinear analysis and applications. Berlin: De Gruyter; 2014.
  • Bacak M. Computing medians and means in Hadamard spaces. SIAM J Optim. 2014;24:1542–1566.
  • Khatibzadeh H. Some remarks on the proximal point algorithm. J Optim Theory Appl. 2012;153:769–778.
  • Khatibzadeh H, Ranjbar S. On the strong convergence of Halpern type proximal point algorithm. J Optim Theory Appl. 2013;158:385–396.
  • Cholamjiak P. The modified proximal point algorithm in CAT(0) spaces. Optim Lett. 2015;9:1401–1410.
  • Dhompongsa S, Panyanak B. On Δ-convergence theorems in CAT(0) spaces. Comput Math Appl. 2008;56:2572–2579.
  • Kirk WA, Panyanak B. A concept of convergence in geodesic spaces. Nonlinear Anal. 2008;68:3689–3696.
  • Halpern B. Fixed point of nonexpanding maps. Bull Am Math Soc. 1967;73:957–961.
  • Saejung S, Yotkaew P. Approximation of zeros of inverse strongly monotone operators in Banach spaces. Nonlinear Anal. 2012;75:742–750.

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.