A multi-step approximant for fixed point problem and convex optimization problem in Hadamard spaces

TL;DR

This paper introduces a multi-step iterative algorithm for solving convex optimization and fixed point problems in Hadamard spaces, providing convergence analysis and extending existing results in the field.

Contribution

It proposes a novel multi-step algorithm with convergence guarantees for convex optimization and fixed point problems in Hadamard spaces, generalizing prior work.

Findings

Established strong and del-convergence of the algorithm.

Computed optimal solutions for convex functions and fixed points.

Extended existing results to broader classes of problems.

Abstract

The purpose of this paper is to propose and analyze a multi-step iterative algorithm to solve a convex optimization problem and a fixed point problem posed on a Hadamard space. The convergence properties of the proposed algorithm are analyzed by employing suitable conditions on the control sequences of parameters and the structural properties of the under lying space. We aim to establish strong and del-convergence results of the proposed iterative algorithm and compute an optimal solution for a minimizer of proper convex lower semicontinuous function and a common fixed point of a finite family of total asymptotically nonexpansive mappings in Hadamard spaces. Our results can be viewed as an extension and generalization of various corresponding results established in the current literature.