A Frank–Wolfe-Type Theorem for Cubic Programs and Solvability for Quadratic Variational Inequalities

Abstract

In this paper, we present a Frank–Wolfe-type theorem for nonconvex cubic programming problems. This result is a direct extension of the previous ones by Andronov et al. (Izvestija Akadem. Nauk SSSR, Tekhnicheskaja Kibernetika 4:194–197, 1982) and Flores-Bazan et al. (Math. Program. 145:263–290, 2014). Under suitable conditions, we characterize the compactness of the solution set of cubic programming problems. Sufficient conditions for the existence of solutions of quadratic variational inequalities are proposed. We also provide several numerical examples, which not only illustrate the obtained results but also show that the existing results cannot apply.