Abstract

Recent characterizations of various types of differentiable generalized monotone maps by Karamardian–Schaible–Crouzeix and their strengthened versions by Crouzeix–Ferland are extended to the nonsmooth case. For nondifferentiable locally Lipschitz maps necessary and/or sufficient conditions for quasimonotonicity, pseudomonotonicity and strict/ strong pseudomonotonicity are derived. To accomplish this, the generalized Jacobian in the sense of Clarke is employed.

Author information

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

Dinh The Luc

Institute of Mathematics, Hanoi, Vietnam.

S. Schaible

Graduate School of Management, University of California, Riverside, CA 92521, USA

Keywords

  • Monotone map
  • quasimonotone map
  • pseudomonotone map
  • generalized Jacobian

References

  1. M. Avriel, W.E. Diewert, S. Schaible and I. Zang: Generalized concavity, Plenum Press, New York and London, 1988.https://doi.org/10.1007/978-1-4684-7600-2Google Scholar
  2. M. Bianchi: Pseudo P-monotone operators and variational inequalities, Working Paper No.6, Istituto di Matematica Generale, Finanziaria ed Econometrica, Universita Cattolica del Sacro Cuore, Milano, 1993.Google Scholar
  3. E. Castagnoli and P. Mazzoleni: Orderings, generalized convexity and monotonicity, in: S. Komlosi, T. Rapcsak, S. Schaible (eds.): Generalized convexity, Springer Verlag, Heidelberg-New York (1994) 250–262.https://doi.org/10.1007/978-3-642-46802-5_19Google Scholar
  4. F.H.Clarke: Optimization and nonsmooth analysis, Wiley, New York, 1983.Google Scholar
  5. R.W. Cottle and J.C. Yao: Pseudomonotone complementarity problems in Hilbert space, Journal of Optimization Theory and Applications, 78 (1992) 281–295.https://doi.org/10.1007/BF00941468Google Scholar
  6. J.P. Crouzeix and J.A. Ferland: Criteria for differentiable generalized monotone maps, Mathematical Programming, 1994 (in press).Google Scholar
  7. R. Ellaia and A. Hassouni: Characterization of nonsmooth functions through their generalized gradients, Optimization 22 (1991) 401-416.https://doi.org/10.1080/02331939108843678Google Scholar
  8. N. Hadjisavvas and S. Schaible: On strong pseudomonotonicity and (semi) strict quasimonotonicity, Journal of Optimization Theory and Applications, 79 (1993) 139–155.https://doi.org/10.1007/BF00941891Google Scholar
  9. N. Hadjisavvas and S. Schaible: Quasimonotone variational inequalities in Banach spaces, Journal of Optimization Theory and Applications, 90, No.1, July 1996 (to appear).https://doi.org/10.1007/BF02192248Google Scholar
  10. P.T. Harker and J.S. Pang: Finite dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications, Mathematical Programming, 48 (1990) 161–220.https://doi.org/10.1007/BF01582255Google Scholar
  11. A. Hassouni: Operateurs quasimonotones; applications à certains problèmes variationnels, Thèse, Mathématiques Appliquées, Université Paul Sabatier, Toulouse, 1993.Google Scholar
  12. J.-B. Hiriart-Urruty, J.-J. Strodiot and V. Hien Nguyen: Generalized Hessian matrix and second order optimality conditions for problems with C 1,1 data, Applied Mathematics and Optimization, 11 (1984) 169–180.https://doi.org/10.1007/BF01442169Google Scholar
  13. S. Karamaradian: Complementarity over cones with monotone and pseudomonotone maps, Journal of Optimization Theory and Applications, 18 (1976) 445–454.https://doi.org/10.1007/BF00932654Google Scholar
  14. S. Karamaradian and S. Schaible: Seven kinds of monotone maps, Journal of Optimization Theory and Applications, 66 (1990) 37–46.https://doi.org/10.1007/BF00940531Google Scholar
  15. S. Karamaradian, S. Schaible and J.P. Crouzeix: Characterizations of generalized monotone maps, Journal of Optimization Theory and Applications, 76 (1993) 399–413.https://doi.org/10.1007/BF00939374Google Scholar
  16. S. Komlosi: On generalized monotonicity of generalized derivatives, in: P.Mazzoleni (ed.): Proceedings of the Workshop on Generalized Concavity for Economic Applications, Pisa, April 1992, Tecnoprint Bologna (1992) 1–7.Google Scholar
  17. S. Komlosi: Generalized monotonicity and generalized convexity, Journal of Optimization Theory and Applications, 84 (1995) 361–376.https://doi.org/10.1007/BF02192119Google Scholar
  18. S. Komlosi: Generalized monotonicity in nonsmooth analysis in: S.Komlosi, T.Rapcsak, S.Schaible (eds.): Generalized convexity, Springer Verlag, Heidelberg-New York (1994) 263–275.Google Scholar
  19. S. Komlosi: Monotonicity and quasimonotonicity in nonsmooth analysis, Department of Mathematics, University of Pisa, 1993.Google Scholar
  20. D.T. Luc: On the maximal monotonicity of subdifferentials, Acta Mathematica Vietnamica 18 (1993) 99–106.Google Scholar
  21. D.T. Luc: Characterizations of quasiconvex functions, Bulletin of the Australian Mathematical Society, 48 (1993) 393–405.https://doi.org/10.1017/S0004972700015859Google Scholar
  22. D.T. Luc: On generalized convex nonsmooth functions, Bulletin of the Australian Mathematical Society, 49 (1994) 139–149.https://doi.org/10.1017/S000497270001618XGoogle Scholar
  23. D.T. Luc: Taylor’s formula for C k,1 functions, SIAM Journal on Optimization, 1994 (in press).https://doi.org/10.1137/0805032Google Scholar
  24. S. Schaible: Generalized monotonicity - a survey, in: S. Komlosi, T. Rapcsak, S. Schaible (eds.): Generalized convexity, Springer Verlag, Heidelberg-New York (1994) 229–249.https://doi.org/10.1007/978-3-642-46802-5_18Google Scholar
  25. S. Schaible and J.C. Yao: On the equivalence of nonlinear complementarity problems and least element problems, Mathematical Programming, 70 (1995) 191–200.https://doi.org/10.1007/BF01585936Google Scholar
  26. J.C. Yao: Variational inequalities with generalized monotone operators, Mathematics of Operations Research, 19 (1994) 691–705.https://doi.org/10.1287/moor.19.3.691Google Scholar
  27. J.C. Yao: Multivalued variational inequalities with K-pseudomonotone operators, Journal of Optimization Theory and Applications, 83, (1994) 391–403.https://doi.org/10.1007/BF02190064Google Scholar
  28. D.L. Zhu and P. Marcotte: New classes of generalized monotonicity, Journal of Optimization Theory and Applications, 87, (1995) 457–471.https://doi.org/10.1007/BF02192574Google Scholar

Suggested citation

Dinh The Luc, S. Schaible. “Generalized Monotone Nonsmooth Maps.” Journal of Convex Analysis 3 (1996), No. 2, 195–205. https://doi.org/10.68381/jca03013

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 1996. Rights now held by Banach Press.