We present some properties of explicitly quasiconvex functions, which play an important role in both multicriteria and scalar optimization. Some of these results were recently obtained together with Ovidiu Bagdasar during a research visit at the University of Derby (UK), within the project

We study semicontinuity of multiple integrals ~ f~,u,Du)dx~ where the vector-valued function u is defined for xE ~C N with values in ~ . The function f(x,s,~) is assumed to be Carath@odory and quasiconvex in Morrey's sense. We give conditions on the growth of f that guarantee the sequential lower semicontinuity of the given integral in the weak topology 1 N of the Sobolev space H 'P(~;~ ). The ...

In this paper we introduce an algorithm for solving variational inequality problems when the operator is pseudomonotone and point-to-set (therefore not relying on continuity assumptions). Our motivation development of a method optimisation appearing in Chebyshev rational generalised approximation problems, where approximations are constructed as ratios linear forms (linear combinations basis fu...

In this paper we propose an inexact proximal point method to solve constrained minimization problems with locally Lipschitz quasiconvex objective functions. Assuming that the function is also bounded from below, lower semicontinuous and using proximal distances, we show that the sequence generated for the method converges to a stationary point of the problem.

We review various sorts of generalized convexity and we raise some questions about them. We stress the importance of some special subclasses of quasiconvex functions. Dedicated to Marc Attéia Résumé. Nous passons en revue quelques notions de convexité généralisée. Nous tentons de les relier et nous soulevons quelques questions. Nous soulignons l’importance de quelques classes particulières de f...

In this paper we introduce the concepts of quasimonotone maps and pseudoconvex functions. Moreover, a notion of pseudomonotonicity for multi mappings is introduced; it is shown that, if a function f is continuous, then its pseudoconvexity is equivalent to the pseudomonotonicity of its generalized subdifferential in the sense of Clarke and Rockafellar and prove that a lower semicontinuous functi...

In this paper we provide some new necessary and sufficient conditions for pseudoconvexity and semistrict quasiconvexity of a given proper extended real-valued function in terms of the Clarke-Rockafellar subdifferential. Further we extend to programs with pseudoconvex objective function two earlier characterizations of the solutions set of a set constrained nonlinear programming problem due to O...