Geodesic
A Characterization of the Solution Set of Pseudoconvex Extremum Problems
Journal of convex analysis, Tome 19 (2012) no. 1, pp. 113-123 Cet article a éte moissonné depuis la source Heldermann Verlag

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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.
Classification : 47H04, 47H05, 90C25
Mots-clés : Paramonotone map, pseudomonotone-star-map, pseudoconvex programs 
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     author = {M. Castellani and M. Giuli},
     title = {A {Characterization} of the {Solution} {Set} of {Pseudoconvex} {Extremum} {Problems}},
     journal = {Journal of convex analysis},
     pages = {113--123},
     year = {2012},
     volume = {19},
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     url = {http://geodesic.mathdoc.fr/item/JCA_2012_19_1_JCA_2012_19_1_a5/}
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M. Castellani; M. Giuli. A Characterization of the Solution Set of Pseudoconvex Extremum Problems. Journal of convex analysis, Tome 19 (2012) no. 1, pp. 113-123. http://geodesic.mathdoc.fr/item/JCA_2012_19_1_JCA_2012_19_1_a5/