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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

Rotundity, smoothness and duality

Penot J.-P.

Control and Cybernetics

2003

Vol. 32, no 4

711-733

The duality between smoothness and rotundity of functions is studied in a nonlinear abstract framework. Here smoothness is enlarged to subdifferentiability properties and rotundity is formulated by means of approximation properties.

Kuhn-Tucker type optimality conditions for some class of nonsmooth programming problems

Śmietański M. J.

Control and Cybernetics

2003

Vol. 32, no 2

361-376

We consider two nonlinear programming problems with nonsmooth functions. The necessary and sufficient first order optimality conditions use the Dini and Clarke derivatives. However, the obtained Kuhn-Tucker conditions have a rather classical form. The sufficient conditions alone are obtained thanks to some properties of generalized convexity and generalized linearity of functions. The necessary and sufficient optimality conditions are given in the Lagrange form.