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2 Journal of Applied Analysis

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On strict pseudoconvexity

Ivanov V., I.

Journal of Applied Analysis

2007

Vol. 13, nr 2

183-196

The present paper provides first and second-order characterizations of a radilly lower semicontinuous strictly pseudoconvex function ∫ : X → R defined on a convex set X in the real Euclidean space Rn in twerms of the lower Dini-directional derivative. In particular we obtain connections between the strictly pseudoconvex functions, nonlinear programming problem, Stampacchia variational inequality, and strict Minty variational inequality. We extend to the radially continuous functions the characterization due to Diewert, Avriel, Zang [6]. A new implication appears in our conditions. Connections with other classes of functions are also derived

Second-order characterizations of convex and pseudoconvex functions

Ginchev I., Ivanov V. I.

Journal of Applied Analysis

2003

Vol. 9, nr 2

261-273

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 [10]. For pseudoconvex functions we generalize results of Diewert, Avriel, Zang [6] and Crouzeix [4]. While some known results on pseudoconvex functions are stated in global concepts (e.g. Komlosi [11]), we succeeded in realizing the task to confine to local concepts only.