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.

中文翻译：

---

三次变分不等式的三次规划的 Frank-Wolfe 型定理和可解性

在本文中，我们提出了非凸三次规划问题的 Frank-Wolfe 型定理。该结果是 Andronov 等人先前结果的直接扩展。(Izvestija Akadem. Nauk SSSR, Tekhnicheskaja Kibernetika 4:194–197, 1982) 和 Flores-Bazan 等人。(Math. Program. 145:263–290, 2014)。在合适的条件下，我们刻画了三次规划问题解集的紧致性。提出了二次变分不等式解存在的充分条件。我们还提供了几个数值例子，它们不仅说明了所获得的结果，而且表明现有的结果是不适用的。