Journal of Convex Analysis 33 (2026), No. 1&2, 455--472
Copyright Heldermann Verlag 2026
The Derivative of a Nonlinear Positively Homogeneous Variational Inequality is a Complementarity Problem
Samir Adly
Institut de Recherche XLIM, UMR CNRS 7252, Université de Limoges, Limoges, France
samir.adly@unilim.fr
Loic Bourdin
Institut de Recherche XLIM, UMR CNRS 7252, Université de Limoges, Limoges, France
loic.bourdin@unilim.fr
Using advanced concepts and techniques from convex and variational analysis (such as the notion of twice epi-differentiability), we establish, under a set of appropriate conditions (including a polyhedricity assumption), that the solution to a parameterized nonlinear variational inequality associated with a positively homogeneous function is differentiable, and furthermore that its derivative is the solution to a corresponding complementarity problem. We illustrate our main result through a series of examples and counterexamples. In particular we introduce an example of a projection operator onto a nonempty closed convex cone that is not directionally differentiable, which is, to our best knowledge, new in the literature.
Keywords: Variational inequalities, complementarity problems, semi-differentiability, twice epi-differentiability, polyhedricity.
MSC: 47J20, 49J40, 90C33, 49J52.
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