Abstract

This paper presents necessary and/or sufficient conditions for a continuous map to be monotone, quasimonotone or pseudomonotone. The results are given in terms of approximate Jacobian matrices which reduce to convexificators for a real-valued map. The results extend corresponding results obtained using Clarke generalized Jacobian matrices.

Author information

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

V. Jeyakumar

Department of Applied Mathematics, University of New South Wales, Sydney 2052, Australia.

D. T. Luc

Institute for Mathematics, Hanoi, Vietnam.

S. Schaible

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

Keywords

  • Monotone maps
  • quasimonotone maps
  • pseudomonotone maps
  • approximate Jacobians

References

  1. M. Bianchi, N. Hadjisavvas, S. Schaible: Vector equilibrium problems with generalized monotone bifunctions, J. Optim. Theor. Appl. 92(3) (1997) 527-542.https://doi.org/10.1023/A:1022603406244Google Scholar
  2. F. H. Clarke: Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, 1983.Google Scholar
  3. J. P. Crouzeix, J. A. Ferland: Criteria for differentiable generalized monotone maps, Math. Progr. 75(3) (1996) 399-406.https://doi.org/10.1007/BF02592191Google Scholar
  4. J. P. Crouzeix, S. Schaible: Generalized monotone affine maps, SIAM J. Matrix Anal. Appl. 17(4) (1996) 992-997.https://doi.org/10.1137/S0895479895290978Google Scholar
  5. V. F. Demyanov, V. Jeyakumar: Hunting for a smaller convex subdifferential, J. Global Optim. 10 (1997) 305-326.https://doi.org/10.1023/A:1008246130864Google Scholar
  6. V. F. Demyanov, A. M. Rubinov: Constructive Nonsmooth Analysis, Verlag Peter Lang, Frankfurt am Main, 1995.Google Scholar
  7. A. Hassouni: Opérateurs Quasimonotone - Applications a Certain Problèmes Variationnels, Thesis, Mathématiques Appliquées, Université Paul Sabatier, Toulouse, France, 1993.Google Scholar
  8. J. B. Hiriart-Urruty: Mean value theorems for vector valued mappings in nonsmooth optimization, Numer. Funct. Anal. Optim. 2 (1980) 1-30.https://doi.org/10.1080/01630568008816042Google Scholar
  9. V. Jeyakumar, D. T. Luc: Nonsmooth calculus, minimality and monotonicity of convexificators, Applied Mathematics Research Report AMR96/29, University of New South Wales, Australia, 1996 (submitted).Google Scholar
  10. V. Jeyakumar, X. Q. Yang: Approximate generalized Hessians and Taylor's expansions for continuously Gâteaux differentiable functions, Applied Mathematics Research Report AMR96/20, University of New South Wales, Australia, 1996 (submitted).Google Scholar
  11. H. Jiang, L. Qi: Local uniqueness and convergence of iterative methods for nonsmooth variational inequalities, J. Math. Anal. Appl. 196 (1995) 314-331.https://doi.org/10.1006/jmaa.1995.1412Google Scholar
  12. S. Karamardian: Complementarity over cones with monotone and pseudomonotone maps, J. Optim. Theor. Appl. 18 (1976) 445-454.https://doi.org/10.1007/BF00932654Google Scholar
  13. S. Karamardian, S. Schaible, J. P. Crouzeix: Characterizations of generalized monotone maps, J. Optim. Theor. Appl. 76 (1993) 399-413.https://doi.org/10.1007/BF00939374Google Scholar
  14. D. T. Luc: Characterizations of quasi-convex functions, Bull. Austr. Math. Soc. 49 (1993) 393-405.https://doi.org/10.1017/S0004972700015859Google Scholar
  15. D. T. Luc, S. Schaible: On generalized monotone nonsmooth maps, J. Convex Analysis 3(2) (1996) 195-205.https://doi.org/10.68381/jca03013Google Scholar
  16. S. Schaible, J. C. Yao: On the equivalence of nonlinear complementarity problems and least element problems, Math. Progr. 70 (1995) 191-200.https://doi.org/10.1007/BF01585936Google Scholar
  17. J. C. Yao: Variational inequalities with generalized monotone operators, Math. Oper. Res. 19 (1994) 691-705.https://doi.org/10.1287/moor.19.3.691Google Scholar

Suggested citation

V. Jeyakumar, D. T. Luc, S. Schaible. “Characterizations of Generalized Monotone Nonsmooth Continuous Maps Using Approximate Jacobians.” Journal of Convex Analysis 5 (1998), No. 1, 119–132. https://doi.org/10.68381/jca05-9

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Published by Heldermann Verlag, 1998. Rights now held by Banach Press LLC.