Оn a proximal point algorithm for solving minimization problem and common fixed point problem in CAT($k$) spaces

  • C. Garodia Jamia Millia Islamia, New Delhi, India
  • S. Radenovic University of Belgrade, Serbia
Keywords: Minimization problem; resolvent operator; CAT(1) space; proximal point algorithm; nonexpansive mapping.


UDC 517.9

We propose a new modified proximal point algorithm in the setting of CAT(1) spaces, which can be used for solving the minimization problem and the  common fixed-point problem.  In addition, we prove several convergence results for the proposed algorithm under certain mild conditions.  Further, we provide some applications for the convex minimization problem and the fixed point problem in the CAT($k$) spaces with a bounded positive real number $k$.  In the process, several relevant results available in the existing literature are generalized and improved.


B. Martinet, Régularisation d'inequations variationnelles par approximations successives, Rev. Fr. Inform. Rech. Op'{e}r., 4, 154–158 (1970). DOI: https://doi.org/10.1051/m2an/197004R301541

R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control and Optim., 14, 877–898 (1976). DOI: https://doi.org/10.1137/0314056

H. Brézis, P. Lions, Produits infinis de résolvantes, Israel J. Math., 29, 329–345 (1978). DOI: https://doi.org/10.1007/BF02761171

O. Güler, On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control and Optim., 29, 403–419 (1991). DOI: https://doi.org/10.1137/0329022

S. Kamimura, W. Takahashi, Approximating solutions of maximal monotone operators in Hilbert spaces, J. Approx. Theory, 106, 226–240 (2000). DOI: https://doi.org/10.1006/jath.2000.3493

M. V. Solodov, B. F. Svaiter, Forcing strong convergence of proximal point iterations in a Hilbert space, Math. Program., 87, 189–202 (2000). DOI: https://doi.org/10.1007/s101079900113

H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66, 240–256 (2002). DOI: https://doi.org/10.1112/S0024610702003332

O. A. Boikanyo, G. Morosanu, A proximal point algorithm converging strongly for general errors, Optim. Lett., 4, 635–641 (2010). DOI: https://doi.org/10.1007/s11590-010-0176-z

S. Kamimura, W. Takahashi, Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim., 13, 938–945 (2002). DOI: https://doi.org/10.1137/S105262340139611X

F. Kohsaka, W. Takahashi, Strong convergence of an iterative sequence for maximal monotone operators in a Banach space, Abstr. and Appl. Anal., 3, 239–249 (2004). DOI: https://doi.org/10.1155/S1085337504309036

K. Aoyama, F. Kohsaka, W. Takahashi, Proximal point methods for monotone operators in banach space, Taiwanese J. Math., 15, № 1, 259–281 (2011). DOI: https://doi.org/10.11650/twjm/1500406174

B. Djafari Rouhani, H. Khatibzadeh, On the proximal point algorithm, J. Optim. Theory and Appl., 137, 411–417 (2008). DOI: https://doi.org/10.1007/s10957-007-9329-3

F. Wang, H. Cui, On the contraction proximal point algorithms with multi parameters, J. Global Optim., 54, 485–491 (2012). DOI: https://doi.org/10.1007/s10898-011-9772-4

O. P. Ferreira, P. R. Oliveira, Proximal point algorithm on Riemannian manifolds, Optimization, 51, 257–270 (2002). DOI: https://doi.org/10.1080/02331930290019413

C. Li, G. López, V. Martín-Márquez, Monotone vector fields and the proximal point algorithm on Hadamard manifolds, J. London Math. Soc., 79, 663–683 (2009). DOI: https://doi.org/10.1112/jlms/jdn087

E. A. Papa Quiroz, P. R. Oliveira, Proximal point methods for quasiconvex and convex functions with Bregman distances on Hadamard manifolds, J. Convex Anal., 16, 49–69 (2009).

J. H. Wang, G. López, Modified proximal point algorithms on Hadamard manifolds, Optimization, 60, 697–708 (2011). DOI: https://doi.org/10.1080/02331934.2010.505962

R. Adler, J. P. Dedieu, J. Y. Margulies, M. Martens, M. Shub, Newton's method on Riemannian manifolds and a geometric model for human spine, IMA J. Numer. Anal., 22, 359–390 (2002). DOI: https://doi.org/10.1093/imanum/22.3.359

S. T. Smith, Optimization techniques on Riemannian manifolds, Fields Institute Communications, vol. 3, Amer. Math. Soc., Providence (1994), p. 113–146. DOI: https://doi.org/10.1090/fic/003/09

C. Udriste, Convex functions and optimization methods on Riemannian manifolds, Math. and Appl., vol. 297. Kluwer Acad., Dordrecht (1994). DOI: https://doi.org/10.1007/978-94-015-8390-9

J. H. Wang, C. Li, Convergence of the family of Euler–Halley type methods on Riemannian manifolds under the $gamma$-condition, Taiwanese J. Math., 13, 585–606 (2009). DOI: https://doi.org/10.11650/twjm/1500405357

M. Bačák, The proximal point algorithm in metric spaces, Israel J. Math., 194, 689–701 (2013). DOI: https://doi.org/10.1007/s11856-012-0091-3

I. Uddin, C. Garodia, S. H. Khan, A proximal point algorithm converging strongly to a minimizer of a convex function, Jordan J. Math. and Stat., 13, № 4, 659–685 (2020).

P. Cholamjiak, A. A. N. Abdou, Y. J. Cho, Proximal point algorithms involving fixed points of nonexpansive mappings in CAT(0) spaces, Fixed Point Theory and Appl., 2015, 1–13 (2015). DOI: https://doi.org/10.1186/s13663-015-0465-4

P. Cholamjiak, The modified proximal point algorithm in CAT(0) spaces, Optim. Lett., 9, 1401–1410 (2015). DOI: https://doi.org/10.1007/s11590-014-0841-8

M. T. Heydari, S. Ranjbar, Halpern-type proximal point algorithm in complete CAT(0) metric spaces, An. Ştiinƫ. Univ. ``Ovidius'' Constanƫa Ser. Mat., 24, № 3, 141–159 (2016). DOI: https://doi.org/10.1515/auom-2016-0052

Y. Kimura, F. Kohsaka, Two modified proximal point algorithms for convex functions in Hadamard spaces, Linear and Nonlinear Anal., 2, 69–86 (2016).

Y. Kimura, F. Kohsaka, The proximal point algorithms in geodesic spaces with curvature bounded above, Linear and Nonlinear Anal., 3, 133–148 (2017).

N. Pakkaranang, P. Kumam, P. Cholamjiak, R. Suparatulatorn, P. Chaipunya, Proximal point algorithms involving fixed point iteration for nonexpansive mappings in CAT(k) spaces, Carpathian J. Math., 34, № 2, 229–237 (2018).

N. Pakkaranang, P. Kumam, C. F. Wen, J. C. Yao, Y. J. Cho, On modified proximal point algorithms for solving minimization problems and fixed point problems in CAT(k) spaces, Math. Methods Appl. Sci. (2019). DOI: https://doi.org/10.1002/mma.5965

N. Wairojjana, P. Saipara, On solving minimization problem and common fixed point problem over geodesic spaces with curvature bounded above, Commun. Math. and Appl., 11, № 3, 443–460 (2020).

Y. Kimura, S. Saejung, P. Yotkaew, The Mann algorithm in a complete geodesic space with curvature bounded above, Fixed Point Theory and Appl., 213, Article ID 336 (2013). DOI: https://doi.org/10.1186/1687-1812-2013-336

Y. Kimura, F. Kohsaka, Spherical nonspreadingness of resolvents of convex functions in geodesic spaces, J. Fixed Point Theory and Appl., 18, 93–115 (2016). DOI: https://doi.org/10.1007/s11784-015-0267-7

B. Panyanak, On total asymptotically nonexpansive mappings in CAT(k) spaces, J. Inequal. and Appl., Article 336 (2014). DOI: https://doi.org/10.1186/1029-242X-2014-336

How to Cite
Garodia, C., and S. Radenovic. “Оn a Proximal Point Algorithm for Solving Minimization Problem and Common Fixed Point Problem in CAT($k$) Spaces”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, no. 2, Mar. 2023, pp. 168 -1, doi:10.37863/umzh.v75i2.6770.
Research articles