i-manager Publications

Strong and Δ - Convergence Theorems using Modified Proximal Point Algorithm in CAt(0) Spaces

Apurva Kumar Das*, Shailesh Dhar Diwan**, Samir Dashputre***

* Lecturer, Department of Mathematics, Government Polytechnic Sukma, Chhattisgarh, India.

** Associate Professor, Department of Mathematics, Government Engineering College, Raipur, Chhattisgarh, India.

*** Assistant Professor, Department of Mathematics, Government College Arjunda, Balod, Chhattisgarh, India.

Periodicity:April - June'2018
DOI : https://doi.org/10.26634/jmat.7.2.13999

Abstract

Let (X, d) be a complete CAT(0) space and f∶ X → (-∞, ∞] be a proper convex and lower semi-continuous function. Suppose T be a nonexpansive mapping on X such that Ω=F(T) ⋂ argmin_(yϵX) f(y) is nonempty. The purpose of this paper is to define a modified proximal point algorithm and prove the existence of a sequence proposed by the authors converges to Ω. In this paper, they prove strong and Δ-convergence theorems with their proposed modified proximal point algorithm in CAT(0) spaces.

Keywords

Strong Convergence,Δ -Convergence, Convex Minimization Problem, Resolvent Identity, CAT(0) Space, Modified Proximal Point Algorithm, Nonexpansive Mapping.

How to Cite this Article?

Das. A.K., Diwan. S.D., and Dashputre. S. (2018). Strong and Δ - Convergence Theorems using Modified Proximal Point Algorithm in CAt(0) Spaces. i-manager’s Journal on Mathematics, 7(2), 8-16. https://doi.org/10.26634/jmat.7.2.13999

References

[1]. Adler, R. L., Dedieu, J. P., Margulies, J. Y., Martens, M., & Shub, M. (2002). Newton's method on Riemannian manifolds and a geometric model for the human spine. IMA Journal of Numerical Analysis, 22(3), 359-390.

[2]. Ambrosio, L., Gigli, N., & Savaré, G. (2008). Gradient Flows: In Metric Spaces and in the Space of Probability Measures. Springer Science & Business Media.

[3]. Ariza-Ruiz, D., Leuştean, L., & López-Acedo, G. (2014). Firmly nonexpansive mappings in classes of geodesic spaces. Transactions of the American Mathematical Society, 366(8), 4299-4322.

[4]. Bačák, M. (2013). The proximal point algorithm in metric spaces. Israel Journal of Mathematics, 194(2), 689-701.

[5]. Bačák, M. (2014). Computing medians and means in Hadamard spaces. SIAM Journal on Optimization, 24(3), 1542- 1566.

[6]. Chang, S. S., Yao, J. C., Wang, L., & Qin, L. J. (2016). Some convergence theorems involving proximal point and common fixed points for asymptotically nonexpansive mappings. Fixed Point Theory and Applications, 2016(1), 68.

[7]. Cho, Y. J., Ćirić, L., & Wang, S. H. (2011). Convergence theorems for nonexpansive semigroups in CAT(0) spaces. Nonlinear Analysis: Theory, Methods & Applications, 74(17), 6050-6059.

[8]. Cholamjiak, P., Abdou, A. A., & Cho, Y. J. (2015). Proximal point algorithms involving fixed points of nonexpansive mappings in CAT (0) spaces. Fixed Point Theory and Applications, 2015(1), 227.

[9]. Dhompongsa, S., & Panyanak, B. (2008). On ∆-convergence theorems in CAT(0) spaces. Computers & Mathematics with Applications, 56(10), 2572-2579.

[10]. Dhompongsa, S., Kirk, W. A., & Sims, B. (2006). Fixed points of uniformly Lipschitzian mappings. Nonlinear Analysis: Theory, Methods & Applications, 65(4), 762-772.

[11]. Ferreira, O. P., & Oliveira, P. R. (2002). Proximal point algorithm on Riemannian manifolds. Optimization, 51(2), 257-270.

[12]. Fukhar-ud-din, H. (2013). Strong convergence of an Ishikawa-type algorithm in CAT(0) spaces. Fixed Point Theory and Applications, 2013, 1-11.

[13]. Ishikawa, S. (1974). Fixed points by a new iteration method. Proceedings of the American Mathematical Society, 44(1), 147-150.

[14]. Jost, J. (1995). Convex functionals and generalized harmonic maps into spaces of non positive curvature. Commentarii Mathematici Helvetici, 70(1), 659-673.

[15]. Kirk, W. A. (2003, February). Geodesic geometry and fixed point theory. In Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003) (Vol. 64, pp. 195-225).

[16]. Laokul, T., & Panyanak, B. (2009). Approximating fixed points of nonexpansive mappings in CAT(0) spaces. International Journal of Mathematical Analysis, 3(27), 1305-1315.

[17]. Li, C., López, G., & Martín-Márquez, V. (2009). Monotone vector fields and the proximal point algorithm on Hadamard manifolds. Journal of the London Mathematical Society, 79(3), 663-683.

[18]. Mann, W. R. (1953). Mean value methods in iteration. Proceedings of the American Mathematical Society, 4(3), 506- 510.

[19]. Martinet, B. (1970). Brève communication. Régularisation d'inéquations variationnelles par approximations successives. Revue française d'informatique et de recherche opérationnelle. Série rouge, 4(R3), 154-158.

[20]. Mayer, U. F. (1998). Gradient flows on nonpositively curved metric spaces and harmonic maps. Communications in Analysis and Geometry, 6(2), 199-253.

[21]. Nanjaras, B., & Panyanak, B. (2010). Demiclosed principle for asymptotically nonexpansive mappings in CAT(0) spaces. Fixed Point Theory and Applications, 2010(1), 268780.

[22]. Osilike, M. O., & Aniagbosor, S. C. (2000). Weak and strong convergence theorems for fixed points of asymptotically nonexpensive mappings. Mathematical and Computer Modelling, 32(10), 1181-1191.

[23]. Quiroz, E. P., & Oliveira, P. R. (2009). Proximal point methods for quasiconvex and convex functions with Bregman distances on Hadamard manifolds. J. Convex Anal., 16(1), 49-69.

[24]. Rockafellar, R. T. (1976). Monotone operators and the proximal point algorithm. SIAM Journal on Control and Optimization, 14(5), 877-898.

[25]. Senter, H. F., & Dotson, W. G. (1974). Approximating fixed points of nonexpansive mappings. Proceedings of the American Mathematical Society, 44(2), 375-380.

[26]. Smith, S. T. (1994). Optimization techniques on Riemannian manifolds. Fields Institute Communications, 3(3), 113-135.

[27]. Udriste, C. (1994). Convex Functions and Optimization Methods on Riemannian Manifolds (Vol. 297). Springer Science & Business Media.

[28]. Wang, J. H., & Li, C. (2009). Convergence of the family of Euler-Halley type methods on Riemannian manifolds under

the g-condition. Taiwanese Journal of Mathematics, 13(2A), 585-606.

[29]. Wang, J. H., & López, G. (2011). Modified proximal point algorithms on Hadamard manifolds. Optimization, 60(6), 697- 708.

If you have access to this article please login to view the article or kindly login to purchase the article

Single Article

	North Americas,UK, Middle East,Europe		India	Rest of world
	USD	EUR	INR	USD-ROW
Online	15	15

ADD TO CART

Options for accessing this content:

If you would like institutional access to this content, please recommend the title to your librarian.
If you already have i-manager's user account: Login above and proceed to purchase the article.
New Users: Please register, then proceed to purchase the article.