In this paper, we deal with a trust-region sequential quadratic programming (SQP) method with a nonsmooth merit function for solving nonlinear optimization poblems. Based on the method proposed by Yamashita and Dan (2005), our method calculates search directions by solving the two types of subproblems, which are a convex QP subproblem and a linear system of equations. When possible, we execute a search along the negative-curvature direction. The proposed method generates negative-curvature directions in the existence of the nonsmooth term in the merit function. We show that the generated sequence converges to a point that satisfies the first-order and second-order optimality conditions.

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, the ham-sandwich theorem, the necklace theorem and coloring of Kneser graph by Lov´asz are well-known. This paper attempts to apply Borsuk-Ulam’s theorem to nonlinear programming.