Second-Order Characterizations of Convex and Pseudoconvex Functions

Abstract

The present paper gives characterizations of radially u.s.c. convex and pseudoconvex functions f : X → ℝ defined on a convex subset X of a real linear space E in terms of first and second-order upper Dini-directional derivatives. Observing that the property f radially u.s.c. does not require a topological structure of E , we draw the possibility to state our results for arbitrary real linear spaces. For convex functions we extend a theorem of Huang, Ng [Math. Oper. Res. 22: 747–753, 1997]. For pseudoconvex functions we generalize results of Diewert, Avriel, Zang [J. Econom. Theory 25: 397–420, 1981] and Crouzeix [Generalized Convexity, Generalized Monotonicity: Recent Results: 237–256, Kluwer Academic Publisher, 1998]. While some known results on pseudoconvex functions are stated in global concepts (e.g. Komlosi [Math. Pro Programming 26: 232–237, 1983]), we succeeded in realizing the task to confine to local concepts only.