Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

Samir Adly

Institut de Recherche XLIM, UMR CNRS 7252, Université de Limoges, Limoges, France

samir.adly@unilim.fr

Loïc Bourdin

Institut de Recherche XLIM, UMR CNRS 7252, Université de Limoges, Limoges, France

loic.bourdin@unilim.fr

Keywords

  • Variational inequalities
  • complementarity problems
  • semi-differentiability
  • twice epi-differentiability
  • polyhedricity

Mathematics Subject Classification

47J20, 49J40, 90C33, 49J52

References

  1. S. Adly, L. Bourdin: Sensitivity analysis of variational inequalities via twice epi-differentiability and proto-differentiability of the proximity operator, SIAM J. Optim. 28/2 (2018) 1699–1725.https://doi.org/10.1137/17M1135013Google Scholar
  2. J. F. Bonnans, A. Shapiro: Perturbation Analysis of Optimization Problems, Springer Series in Operations Research, Springer, New York (2000).https://doi.org/10.1007/978-1-4612-1394-9Google Scholar
  3. H. Brézis: Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier (Grenoble) 18 (1968) 115–175.https://doi.org/10.5802/aif.280Google Scholar
  4. H. Brézis: Problèmes unilatéraux, J. Math. Pures Appl. 51 (1972) 1–168.Google Scholar
  5. H. Brézis: Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland Mathematics Studies Vol. 5, North-Holland, Amsterdam (1973).Google Scholar
  6. F. Browder: Nonlinear maximal monotone mappings in Banach spaces, Math. Annalen 175 (1968) 81–113.https://doi.org/10.1007/BF01418765Google Scholar
  7. C. Christof: Sensitivity Analysis of Elliptic Variational Inequalities of the First and the Second Kind, PhD thesis, Technische Universität Dortmund (2019).Google Scholar
  8. C. Christof, G. Wachsmuth: On the non-polyhedricity of sets with upper and lower bounds in dual spaces, GAMM-Mitteilungen 40/4 (2018) 339–350.https://doi.org/10.1002/gamm.201740005Google Scholar
  9. C. Christof, G. Wachsmuth: Differential sensitivity analysis of variational inequalities with locally Lipschitz continuous solution operators, Appl. Math. Optim. 81/1 (2020) 23–62.https://doi.org/10.1007/s00245-018-09553-yGoogle Scholar
  10. C. N. Do: Generalized second-order derivatives of convex functions in reflexive Banach spaces, Trans. Amer. Math. Soc. 334/1 (1992) 281–301.https://doi.org/10.1090/S0002-9947-1992-1088019-1Google Scholar
  11. M. C. Ferris, J. S. Pang: Engineering and economic applications of complementarity problems, SIAM Review 39 (1997) 669–713.https://doi.org/10.1137/S0036144595285963Google Scholar
  12. G. Fichera: Problemi elastostatici con vincoli unilaterali: Il problema di signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. 35 (1963) 1061–1085.Google Scholar
  13. A. Haraux: How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities, J. Math. Soc. Japan 29/4 (1977) 615–631.https://doi.org/10.2969/jmsj/02940615Google Scholar
  14. M. Hintermüller, T. M. Surowiec: On the directional differentiability of the solution mapping for a class of variational inequalities of the second kind, Set-Valued Var. Analysis 26/3 (2018) 631–642.https://doi.org/10.1007/s11228-017-0408-9Google Scholar
  15. J. B. Kruskal: Two convex counterexamples: A discontinuous envelope function and a nondifferentiable nearest-point mapping, Proc. Amer. Math. Soc. 23 (1969) 697–703.https://doi.org/10.1090/S0002-9939-1969-0259752-9Google Scholar
  16. J.-L. Lions: Quelques Méthodes de Résolution des Problèmes aux Limites Non-Linéaires, Dunod, Paris (1969).Google Scholar
  17. F. Mignot: Contrôle dans les inéquations variationelles elliptiques, J. Funct. Analysis 22/2 (1976) 130–185.https://doi.org/10.1016/0022-1236(76)90017-3Google Scholar
  18. B. Mordukhovich, N. M. Nam: An Easy Path to Convex Analysis and Applications, 2nd ed., Synthesis Lectures on Mathematics and Statistics, Springer, Cham (2023).https://doi.org/10.1007/978-3-031-26458-0Google Scholar
  19. J. J. Moreau: Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France 93 (1965) 273–299.https://doi.org/10.24033/bsmf.1625Google Scholar
  20. U. Mosco: A remark on a theorem of F. E. Browder, J. Math. Analysis Appl. 20 (1967) 90–93.https://doi.org/10.1016/0022-247X(67)90108-4Google Scholar
  21. N. Parikh, S. Boyd: Proximal algorithms, Found. Trends Optim. 1/3 (2014) 127–-239.https://doi.org/10.1561/2400000003Google Scholar
  22. R. T. Rockafellar: On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970) 209–216.https://doi.org/10.2140/pjm.1970.33.209Google Scholar
  23. R. T. Rockafellar: Maximal monotone relations and the second derivatives of nonsmooth functions, Ann. Inst. H. Poincaré Anal. Non Linéaire 2/3 (1985) 167–184.https://doi.org/10.1016/s0294-1449(16)30401-2Google Scholar
  24. R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Grundlehren der Mathematischen Wissenschaften Vol. 317, Springer, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3Google Scholar
  25. A. Shapiro: Directionally nondifferentiable metric projection, J. Optim. Theory Appl. 81/1 (1994) 203–204.https://doi.org/10.1007/BF02190320Google Scholar
  26. G. Stampacchia: Formes bilinéaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris 258 (1964) 4413–4416.Google Scholar
  27. G. Stampacchia: Variational Inequalities, Theory and Applications of Monotone Operators, Edizioni Cremonese, Rome (1969).Google Scholar
  28. G. Wachsmuth: A guided tour of polyhedric sets: basic properties, new results on intersections, and applications, J. Convex Analysis 26/1 (2019) 153–188.https://doi.org/10.68381/jca26010Google Scholar

Suggested citation

S. Adly, L. Bourdin. “The Derivative of a Nonlinear Positively Homogeneous Variational Inequality is a Complementarity Problem.” Journal of Convex Analysis 33 (2026), No. 1&2, 455–472. https://doi.org/10.68381/jca33029

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2026. Rights now held by Banach Press LLC.