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

Kallol Paul, Saikat Roy, Debmalya Sain, Shamim Sohel

TL;DR

This paper develops a unified, orthogonality-driven framework to solve key optimization problems in Banach spaces by casting them as specialized least-squares problems in appropriate sequence spaces. Using Birkhoff-James orthogonality, it establishes a duality between the Fermat-Torricelli problem and the Chebyshev center problem, with Fermat-Torricelli corresponding to $\ell_1^n$ and Chebyshev center to $\ell_\infty^n$ formulations, and leverages this to obtain explicit, constructive solutions. It provides complete classifications of BJ-orthogonality in $\ell_1^3$ and $\ell_1^4$ to solve weighted Fermat-Torricelli problems for 3 and 4 points in the complex plane, and derives algorithmic and geometric criteria for the Chebyshev center in the Euclidean plane, including weighted variants and coincidence conditions. Collectively, the results offer a cohesive, duality-based methodology for tackling fundamental location and best-approximation problems in finite-dimensional Banach spaces, with clear geometric interpretations and potential extensions to broader contexts.

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.

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

TL;DR

This paper develops a unified, orthogonality-driven framework to solve key optimization problems in Banach spaces by casting them as specialized least-squares problems in appropriate sequence spaces. Using Birkhoff-James orthogonality, it establishes a duality between the Fermat-Torricelli problem and the Chebyshev center problem, with Fermat-Torricelli corresponding to and Chebyshev center to formulations, and leverages this to obtain explicit, constructive solutions. It provides complete classifications of BJ-orthogonality in and to solve weighted Fermat-Torricelli problems for 3 and 4 points in the complex plane, and derives algorithmic and geometric criteria for the Chebyshev center in the Euclidean plane, including weighted variants and coincidence conditions. Collectively, the results offer a cohesive, duality-based methodology for tackling fundamental location and best-approximation problems in finite-dimensional Banach spaces, with clear geometric interpretations and potential extensions to broader contexts.

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.
Paper Structure (4 sections, 26 theorems, 70 equations)

This paper contains 4 sections, 26 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Connection between the least square problem, the Fermat-Torricelli problem and the Chebyshev center problem
  3. Main Results.
  4. Section -II

Key Result

Theorem 1.1

J Let $\mathbb{X}$ be a Banach space and let $x, y \in \mathbb{X}.$ Then $x \perp_B y$ if and only if there exists $f \in J(x)$ such that $f(y)=0.$

Theorems & Definitions (49)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 39 more