This paper presents a new application of Borsuk-Ulam's theorem to nonlinear programming

TL;DR

This paper explores a novel application of Borsuk-Ulam's theorem from algebraic topology to the field of nonlinear programming, aiming to provide new insights and tools for optimization problems.

Contribution

It introduces a new approach by applying Borsuk-Ulam's theorem to nonlinear programming, bridging topology and optimization in a novel way.

Findings

Establishes a theoretical connection between topology and nonlinear programming.

Provides a framework for potential new solution methods in nonlinear optimization.

Lays groundwork for future research integrating algebraic topology into optimization theory.

Abstract

Borsuk-Ulam's theorem is a useful tool of algebraic topology. It states that for any continuous mapping $f$ from the $n$-sphere to the $n$-dimensional Euclidean space, there exists a pair of antipodal points such that $f(x)=f(-x)$. As for its applications, ham-sandwich theorem, necklace theorem and coloring of Kneser graph by Lov\'{a}sz are well-known. This paper attempts to apply Borsuk-Ulam's theorem to nonlinear programming.