Convergence of proximal splitting algorithms in $\operatorname{CAT}(\kappa)$ spaces and beyond

Fixed Point Theory and Applications Pub Date : 2021-08-24 DOI:10.1186/s13663-021-00698-0

Lauster, Florian, Luke, D. Russell

引用次数: 1

Abstract

In the setting of $\operatorname{CAT}(\kappa)$ spaces, common fixed point iterations built from prox mappings (e.g. prox-prox, Krasnoselsky–Mann relaxations, nonlinear projected-gradients) converge locally linearly under the assumption of linear metric subregularity. Linear metric subregularity is in any case necessary for linearly convergent fixed point sequences, so the result is tight. To show this, we develop a theory of fixed point mappings that violate the usual assumptions of nonexpansiveness and firm nonexpansiveness in p-uniformly convex spaces.

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

近端分裂算法在$\operatorname{CAT}(\kappa)$空间及以后的收敛性

在$\operatorname{CAT}(\kappa)$空间中，由prox映射构建的公共不动点迭代(如prox-prox, Krasnoselsky-Mann松弛，非线性投影梯度)在线性度量子正则性的假设下局部线性收敛。对于线性收敛的不动点序列，线性度量子正则性在任何情况下都是必要的，因此结果是严密的。为了证明这一点，我们建立了一个不动点映射的理论，该理论违背了p-一致凸空间中通常的非扩张性和坚定非扩张性假设。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文去求助

来源期刊

Fixed Point Theory and Applications MATHEMATICS, APPLIED-MATHEMATICS

自引率

0.00%

发文量

期刊介绍： In a wide range of mathematical, computational, economical, modeling and engineering problems, the existence of a solution to a theoretical or real world problem is equivalent to the existence of a fixed point for a suitable map or operator. Fixed points are therefore of paramount importance in many areas of mathematics, sciences and engineering. The theory itself is a beautiful mixture of analysis (pure and applied), topology and geometry. Over the last 60 years or so, the theory of fixed points has been revealed as a very powerful and important tool in the study of nonlinear phenomena. In particular, fixed point techniques have been applied in such diverse fields as biology, chemistry, physics, engineering, game theory and economics. In numerous cases finding the exact solution is not possible; hence it is necessary to develop appropriate algorithms to approximate the requested result. This is strongly related to control and optimization problems arising in the different sciences and in engineering problems. Many situations in the study of nonlinear equations, calculus of variations, partial differential equations, optimal control and inverse problems can be formulated in terms of fixed point problems or optimization.

Convergence of proximal splitting algorithms in \(\operatorname{CAT}(\kappa)\) spaces and beyond

Abstract