New vectorial versions of Takahashi's nonconvex minimization problem

In this article, some new vectorial versions of Takahashi's nonconvex minimization theorem, which involve algebraic notions instead of topological notions, are established. A nonlinear separation theorem, which extends the result derived by Gerth and Weidner (JAMA 67:297-320, 1990) to general linear spaces (not necessarily endowed with a topology), is proved. Some examples, in order to illustrate and compare the results of this article with the corresponding known results from the literature, are provided.