KaydedilenlerDerginizi ekleyin

Dergiler / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 49 Issue: 1

The strong convergence of a proximal point algorithm in complete CAT(0) metric spaces

Mohsen Tahernia(Arak University)0000-0002-0014-1052
Sirous Moradi(Arak University)0000-0002-8640-7252
Somayeh Jafari(Arak University)0000-0002-3832-3177
Sayfa
—
Yayın Tarihi
2020-02-06
🏛️Bu makalenin bibliyografik verileri ve tam metin bağlantısı DergiPark üzerinden sağlanmıştır.
DergiPark Kaydı ↗
✨

Yapay Zeka Hızlı Bakış & Çıkarımlar

AI Insight
⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

Afterwards, we prove that, under suitable conditions, the proposed algorithm converges strongly to the metric projection of some point onto the zero set of the involving maximal monotone operator.

📌 3 Maddede Bu Makale

🎯
Amaç & Hipotez:In this paper, we consider a proximal point algorithm for finding zeros of maximal monotone operators in complete CAT(0) spaces.
📊
Temel Bulgular & Veri:First, a necessary and sufficient condition is presented for the zero set of the operator to be nonempty.
💡
Sonuç & Katkı:Afterwards, we prove that, under suitable conditions, the proposed algorithm converges strongly to the metric projection of some point onto the zero set of the involving maximal monotone operator.
🔬Tür:Research Article
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Maximal monotone operator · proximal point algorithm · Hadamard space

Abstract

In this paper, we consider a proximal point algorithm for finding zeros of maximal monotone operators in complete CAT(0) spaces. First, a necessary and sufficient condition is presented for the zero set of the operator to be nonempty. Afterwards, we prove that, under suitable conditions, the proposed algorithm converges strongly to the metric projection of some point onto the zero set of the involving maximal monotone operator.

🔖

Bu Makaleye Atıf Yap

Tahernia, M., Moradi, S., Jafari, S. (2020). The strong convergence of a proximal point algorithm in complete CAT(0) metric spaces. Hacettepe Journal of Mathematics and Statistics, Volume: 49 Issue: 1. https://doi.org/10.15672/hujms.470975

📄 Tam Metin PDF Okuyucu

📄

PDF görüntüleyici hazırlanıyor...

Kaynakça (26)

  1. P. Ahmadi and H. Khatibzadeh, On the convergence of inexact proximal point algorithm on Hadamard manifolds, Taiwanese J. Math. 18, 419–433, 2014.
  2. B. Ahmadi Kakavandi, Weak topologies in complete CAT(0) metric spaces, Proc. Amer. Math. Soc. 141, 1029–1039, 2013.
  3. B. Ahmadi Kakavandi and M. Amini, Duality and subdifierential for convex functions on complete CAT(0) metric spaces, Nonlinear Anal. 73, 3450–3455, 2010.
  4. M. Bacak, Convex analysis and optimization in Hadamard spaces, De Gruyter Series in Nonlinear Analysis and Applications, 22, De Gruyter, Berlin, 2014.
  5. I.D. Berg and I.G. Nikolaev, Quasilinearization and curvature of Alexandrov spaces, Geom. Dedicata, 133, 195–218, 2008.
  6. H. Br´ezis and P.L. Lions, Produits infinis de r´esolvantes, Israel J. Math. 29, 329–345, 1978.
  7. M. Bridson and A. Haefliger, Metric spaces of non-positive curvature, 319, Springer, Berlin, 1999.
  8. K.S. Brown, Buildings, Springer, New York, 1989.
  9. D. Burago, Y. Burago and S. Ivanov, A course in metric geometry, Graduate Studies in Mathematics, 33 American Mathematical Society, Providence, RI, 2001.
  10. H. Dehghan and J. Rooin, Metric projection and convergence theorems for nonexpansive mappings in Hadamard spaces, arXiv: 1410.1137v1[math.FA].
  11. S. Dhompongsa and B. Panyanak, On △-convergence theorems in CAT(0) spaces, Comput. Math. Appl. 56, 2572–2579, 2008.
  12. B. Djafari Rouhani and H. Khatibzadeh, On the proximal point algorithm, J. Optim. Theory Appl. 137, 411–417, 2008.
  13. B. Djafari Rouhani and S. Moradi, Strong convergence of two proximal point algorithms with possible unbounded error sequences, J. Optim.Theory Appl. 172, 222–235, 2017.
  14. R. Esp´inola and A. Fern´andez-Le´on, CAT(k)-spaces, weak convergence and fixed points, J. Math. Anal. Appl. 353, 410–427, 2009.
  15. K. Goebel and S. Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings. Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker Inc., New York, 1984.
  16. M. Gromov and S.M. Bates, Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics, 152, eds. J. Lafontaine and P. Pansu) (Birkh¨auser, Boston, 1999, with appendices by M. Katz, P. Pansu and S. Semmes.
  17. O. G¨uler, On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Optim. 29, 403–419, 1991.
  18. M.T. Heydari and S. Ranjbar, Halpern-type proximal point algorithm in complete CAT(0) metric spaces, An. St. Univ. Ovidius Constanta. 24 (3), 141–159, 2016.
  19. J. Jöst, Nonpositive curvature: geometric and analytic aspects, Lectures in Mathematics, Birkhauser, Basel, 1997.
  20. H. Khatibzadeh and S. Ranjbar, Monotone operators and the proximal point algorithm in complete CAT(0) metric spaces, J. Aust. Math. Soc. 103 (1), 70–90, 2017.
  21. W.A. Kirk, Fixed point theorems in CAT(0) spaces and $\mathbb{R}$-trees, J. Fixed Point Theory Appl. 4, 309–316, 2004.
  22. W.A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal. (TMA) 68, 3689–3696, 2008.
  23. C. Li, G. Lopez and V. Martin-Marquez, Monotone vector fields and the proximal point algorithm on Hadamard manifolds, J. London Math. Soc. 79 (2), 663–683, 2009.
  24. T.C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60, 179– 182, 1976.
  25. B. Martinet, Régularisation d,inéquations variationnelles par approximations successives, Rev. Fran´caise d,Inform. et de Rech. Opérationnelle, 3, 154–158, 1970.
  26. R.T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14, 877–898, 1976.