Journal of Convex Analysis 19 (2012), No. 1, 113--123
Copyright Heldermann Verlag 2012
A Characterization of the Solution Set of Pseudoconvex Extremum Problems
Marco Castellani
Dip. Sistemi ed Istituzioni per l'Economia, University of L'Aquila, Struttura Reiss Romoli, Via Giovanni Falcone 25, 67100 L'Aquila, Italy
Massimiliano Giuli
Dip. Sistemi ed Istituzioni per l'Economia, University of L'Aquila, Struttura Reiss Romoli, Via Giovanni Falcone 25, 67100 L'Aquila, Italy
massimiliano.giuli@univaq.it
Pseudomonotone* single--valued functions were introduced by J.-P. Crouzeix, P. Marcotte and D. Zhu [Conditions ensuring the applicability of cutting-plane methods for solving variational inequalities, Mathematical Programming 88 (2000) 521--539] and it was proved that the gradient of a differentiable pseudoconvex function is pseudomonotone*. In the same paper this concept was extended in a natural way to multivalued maps but, to date, there is no result that relates multivalued pseudomonotone* maps to the subdifferential of locally Lipschitz pseudoconvex functions. In this paper, we give a nonsmooth Lipschitz pseudoconvex function whose subdifferential is not pseudomonotone* in the sense of the paper cited above. Besides such a characterization was achieved by N. Hadjisavvas and S. Schaible [On a generalization of paramonotone maps and its application to solving the Stampacchia variational inequality, Optimization 55 (2006) 593--604] using a weaker definition of pseudomonotonicity*. Exploiting this weaker concept, we provide a characterization of the solution set of pseudoconvex programs.
Keywords: Paramonotone map, pseudomonotone-star-map, pseudoconvex programs.
MSC: 47H04, 47H05, 90C25
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