A Theorem of the Alternative and a Two-function Minimax Theorem

“Theorem of the alternative” is the generic name of different results used as an important tool in optimization. Though the early versions have been obtained under strong conditions of convexity, for the further extensions, these conditions have been weakened using generalized convexity beyond vector space structure. This is the case of the result proved by Jeyakumar [4], who used an extension of Fan’s convexlike property. Our theorem of the alternative, proved in Section 3, extends the cited result in the case of the usual order of the n-dimensional Euclidean space, and makes use of the weakened convexlike property introduced in Section 2. The two-function minimax theorem extends the classical minimax inequality to the case of two functions. It is motivated both by the equilibrium problem in the noncooperative game theory and by the generalized duality in optimization. The first known result is due to Fan [3]. Several authors, especially in the last two decades, produced different versions of this theorem, employing the basic techniques from single-function minimax theory. We refer the reader to Simons [9] for a valuable survey in this framework. New results, based mainly on special connectedness properties, are obtained by Kindler in [5]. The main theorem of Section 4 is based on the theorem of the alternative proved in Section 3. For both results, one requires neither vector space structures nor topological support.