The proximal point algorithm in complete geodesic spaces with negative curvature

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

1. M. Ba¥v{c}¥'{a}k, ¥emph{The proximal point algorithm in metric spaces}, Isreal J. Math. ¥textbf{29} (2013), 689--701.
2. M. Bacak, Convex Analysis and Optimization in Hadamard Spaces, De Gruyter, Wurzbrung, 2014.
3. M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer-Verlag, Berlin, 1999.
4. S. Dhompongsa, W. A. Kirk, B. Sims, Fixed points of uniformly lipschitzian mappings, Nonlinear Analysis. 65 (2006), 762--772.
5. T. Kajimura and Y. Kimura, Resolvents of convex functions in complete geodesic spaces with negative curvature, J. Fixed Point Theory Appl. 21 (2019).
6. Y. Kimura and F. Kohsaka, Spherical nonspreadingness of resolvents of convex functions in geodesic spaces, J. Fixed Point Theory Appl. 18 (2016), 93--115.
7. 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.
8. F. Kohsaka, Existence and approximation of fixed points of vicinal mappings in geodesic spaces, Pure Appl. Funct. Anal. 3 (2018), 91--106.

9. U. F. Mayer, Gradient flows on nonpositively curved metric spaces and harmonic maps, Comm. Anal. Geom. 6 (1998), 199--206.
10. R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14 (1976), 877--898.