arXiv subjects
Publications and source records attributed to Fumiaki Kohsaka.
We prove existence and convergence of sequences generated by the proximal point method and its two variants for monotone vector fields in Hadamard spaces. Before obtaining our results, we investigate some fundamental properties of tangent spaces, resolvents, and monotone vector fields in such spaces.
We obtain Sion's minimax theorem in Hadamard spaces and discuss its applications. Among other things, we study several fundamental properties of resolvents of saddle functions in Hadamard spaces. An application to the proximal point algorithm for minimax problems in Hadamard spaces are also included.
We study the existence and approximation of fixed points of metrically nonspreading mappings and firmly metrically nonspreading mappings in Hadamard spaces. The resolvents of monotone operators satisfying range conditions are typical examples of firmly metrically nonspreading mappings. Applications to monotone operators in such spaces are also included.
In this paper, we discuss about monotone vector fields, which is a typical extension to the theory of convex functions, by exploiting the tangent space structure. This new approach to monotonicity in CAT(0) spaces stands in opposed to the monotonicity defined earlier in CAT(0) spaces by Khatibzadeh and Ranjbar [14] and Chaipunya and Kumam [8]. In particular, this new concept extends the theory from both Hilbert spaces and Hadamard manifolds, while the known concept barely has any obvious relationship to the theory in Hadamard manifolds. We also study the corresponding resolvents and Yosida approximations of a given monotone vector field and derive many of their important properties. Finally, we prove a generation theorem by showing convergence of an exponential formula applied to resolvents of a monotone vector field. Our findings improve several known results in the literature including generation theorems of Jost [13, Theorem 1.3.13], Mayer [19, Theorem 1.13], Stojkovic [22, Theorem 2.18], and Bacák [4, Theorem 1.5] for proper, convex, lower semicontinuous functions in the context of complete CAT(0) spaces, and also by Iwamiya and Okochi [11, Theorem 4.1] for monotone vector fields in the context of Hadamard manifolds.
We obtain existence and convergence theorems on two variants of the proximal point algorithm for proper lower semicontinuous convex functions in complete geodesic spaces with curvature bounded above.
We propose the concepts of vicinal mappings and firmly vicinal mappings in metric spaces. We obtain fixed point and convergence theorems for these mappings in complete geodesic spaces with curvature bounded above by one and apply our results to convex optimization in such spaces.
We investigate the asymptotic behavior of sequences generated by the proximal point algorithm for convex functions in complete geodesic spaces with curvature bounded above. Using the notion of resolvents of such functions, which was recently introduced by the authors, we show the existence of minimizers of convex functions under the boundedness assumptions on such sequences as well as the convergence of such sequences to minimizers of given functions.
The aim of this paper is to establish strong convergence theorems for a strongly relatively nonexpansive sequence in a smooth and uniformly convex Banach space. Then we employ our results to approximate solutions of the zero point problem for a maximal monotone operator and the fixed point problem for a relatively nonexpansive mapping.