A Characterization of the Solution Set of Pseudoconvex Extremum Problems

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.