CONVEX-LIKE AND CONCAVE-LIKE CONDITIONS IN ALTERNATIVE, MINIMAX, AND MINIMIZATION THEOREMS

Convexlike and concavelike conditions are of interest for extensions of the Von Neumann minimax theorem. Since the beginning of the 80's, these conditions also play a certain role in deriving generalized alternative theorems of the Gordan, Motzkin, and Farkas type and Lagrange multiplier results for constrained minimization problems. In this paper, we study various known convexlike conditions for vector-valued functions on a set GAMMA and investigate convexlike and concavelike conditions for real-valued functions on a product set C x D, where we are mainly interested in the relationships between these conditions. At the end of the paper, we point out several conclusions from our results for the above-mentioned mathematical fields.