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

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.