The proximal point algorithm in complete geodesic spaces with negative curvature

Yıl 2019, Cilt: 3 Sayı: 4, 192 - 200, 30.12.2019

Takuto Kajimura Yasunori Kimura

https://doi.org/10.31197/atnaa.573972

Cited By: 1

Öz

The proximal point algorithm is an approximation method for finding a minimizer of a convex function. In this paper, we introduce the resolvent for a convex function in complete geodesic spaces with negative curvature. Using properties of the resolvent, we show the proximal point algorithm in complete geodesic spaces with negative curvature. 

Anahtar Kelimeler

CAT($-1$) space, proximal point algorithm, resolvent, convex function

Kaynakça

• M. Ba¥v{c}¥'{a}k, ¥emph{The proximal point algorithm in metric spaces}, Isreal J. Math. ¥textbf{29} (2013), 689--701.
• M. Bacak, Convex Analysis and Optimization in Hadamard Spaces, De Gruyter, Wurzbrung, 2014.
• M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer-Verlag, Berlin, 1999.
• S. Dhompongsa, W. A. Kirk, B. Sims, Fixed points of uniformly lipschitzian mappings, Nonlinear Analysis. 65 (2006), 762--772.
• T. Kajimura and Y. Kimura, Resolvents of convex functions in complete geodesic spaces with negative curvature, J. Fixed Point Theory Appl. 21 (2019).
• Y. Kimura and F. Kohsaka, Spherical nonspreadingness of resolvents of convex functions in geodesic spaces, J. Fixed Point Theory Appl. 18 (2016), 93--115.
• Y. Kimura and F. Kohsaka, The proximal point algorithm in geodesic spaces with curvature bounded above, Linear and Nonlinear Analysis 3, No. 1 (2017), 73--86.
• F. Kohsaka, Existence and approximation of fixed points of vicinal mappings in geodesic spaces, Pure Appl. Funct. Anal. 3 (2018), 91--106.
• U. F. Mayer, Gradient flows on nonpositively curved metric spaces and harmonic maps, Comm. Anal. Geom. 6 (1998), 199--206.
• R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14 (1976), 877--898.