Abstract

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.

Author information

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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

Keywords

  • Paramonotone map
  • pseudomonotone-star-map
  • pseudoconvex programs

Mathematics Subject Classification

47H04, 47H05, 90C25

References

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Suggested citation

M. Castellani, M. Giuli. “A Characterization of the Solution Set of Pseudoconvex Extremum Problems.” Journal of Convex Analysis 19 (2012), No. 1, 113–123. https://doi.org/10.68381/jca19006

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Published by Heldermann Verlag, 2012. Rights now held by Banach Press.