A unified approach to a family of optimization problems in Banach spaces

TL;DR

This paper introduces a unified method using Birkhoff-James orthogonality to solve various optimization problems in Banach spaces, including Fermat-Torricelli and Chebyshev center problems, highlighting their duality and behavior under point modifications.

Contribution

It presents a novel unified approach leveraging Birkhoff-James orthogonality to analyze and solve multiple optimization problems in Banach spaces, establishing duality and new results.

Findings

Duality between Fermat-Torricelli and Chebyshev center problems

Solutions for Fermat-Torricelli problem with three and four points

Behavior of Fermat-Torricelli points under point addition or replacement

Abstract

Our principal aim is to illustrate that the concept Birkhoff-James orthogonality can be applied effectively to obtain a unified approach to a large family of optimization problems in Banach spaces. We study such optimization problems from the perspective of Birkhoff-James orthogonality in certain suitable Banach spaces. In particular, we demonstrate the duality between the Fermat-Torricelli problem and the Chebyshev center problem which are important particular cases of the least square problem. We revisit the Fermat-Torricelli problem for three and four points and solve it using the same technique. We also investigate the behavior of the Fermat-Torricelli points under the addition or replacement of a new point, and present several new results involving the locations of the Fermat-Torricelli point and the Chebyshev center.