A Modified Proximal Point Algorithm and Some Convergence Results

Shengquan Weng,Dingping Wu,Zengfu Chao,Jen-Chih Yao

doi:10.1155/2021/6951062

Shengquan Weng, Dingping Wu + Show 2 more

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https://doi.org/10.1155/2021/6951062

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Abstract

In this paper, the convergence to minimizers of a convex function of a modified proximal point algorithm involving a single-valued nonexpansive mapping and a multivalued nonexpansive mapping in CAT(0) spaces is studied and a numerical example is given to support our main results.

Highlights

In daily life, no matter what we do, there are always many options available and many possible outcomes
0, M2κ is 2-sphere S2 with the the Euclidean plane; if κ > 0, the√n M2κ is metric scaled by a factor (1/ κ )
Let Δ be a geodesic triangle in X with a perimeter less than 2Dκ

Summary

Introduction

No matter what we do, there are always many options available and many possible outcomes. In 1976, in a Hilbert space, Rockafellar [14] studied the convergence to a solution of the convex minimization problem by the proximal point algorithm and proved and obtained a main conclusion that the sequence 􏼈xn􏼉 converges weakly to a minimizer of a convex function f such that 􏽐∞ n 1 λn ∞. In 2013, the proximal point algorithm was introduced by Bacak [15] into CAT(0) spaces (X, d) as follows: x1 ∈ X, for each n ∈ N, xn+1. The proximal point algorithm has been combined with many iterative methods, and a new construction algorithm is further proposed to find approximating fixed points of nonlinear mappings and a proper convex lower semicontinuous function f. Proposed a proximal point algorithm for a hybrid pair of nonexpansive single-valued and multivalued mappings in geodesic metric spaces as follows:. The convergence of the proposed algorithm is studied and its convergence analysis in the end is given

Preliminaries

Main Results

Numerical Experiments