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Characterization of Nonsmooth Semistrictly Quasiconvex and Strictly Quasiconvex Functions

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  • A. Daniilidis

    (University of the Aegean)

  • N. Hadjisavvas

    (University of the Aegean)

Abstract

New concepts of semistrict quasimonotonicity and strict quasimonotonicity for multivalued maps are introduced. It is shown that a locally Lipschitz map is (semi)strictly quasiconvex if and only if its Clarke subdifferential is (semi)strictly quasimonotone. Finally, an existence result for the corresponding variational inequality problem is obtained.

Suggested Citation

  • A. Daniilidis & N. Hadjisavvas, 1999. "Characterization of Nonsmooth Semistrictly Quasiconvex and Strictly Quasiconvex Functions," Journal of Optimization Theory and Applications, Springer, vol. 102(3), pages 525-536, September.
    • Handle: RePEc:spr:joptap:v:102:y:1999:i:3:d:10.1023_a:1022693822102
    DOI: 10.1023/A:1022693822102
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    References listed on IDEAS

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    1. A. Daniilidis & N. Hadjisavvas & S. Schaible, 1997. "Connectedness of the Efficient Set for Three-Objective Quasiconcave Maximization Problems," Journal of Optimization Theory and Applications, Springer, vol. 93(3), pages 517-524, June.
    2. D. Aussel, 1998. "Subdifferential Properties of Quasiconvex and Pseudoconvex Functions: Unified Approach," Journal of Optimization Theory and Applications, Springer, vol. 97(1), pages 29-45, April.
    3. J. Benoist, 1998. "Connectedness of the Efficient Set for Strictly Quasiconcave Sets," Journal of Optimization Theory and Applications, Springer, vol. 96(3), pages 627-654, March.
    4. I. V. Konnov, 1998. "On Quasimonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 99(1), pages 165-181, October.
    5. J.-P. Penot & P. H. Quang, 1997. "Generalized Convexity of Functions and Generalized Monotonicity of Set-Valued Maps," Journal of Optimization Theory and Applications, Springer, vol. 92(2), pages 343-356, February.
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    Citations

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    Cited by:

    1. Vsevolod Ivanov, 2010. "On a theorem due to Crouzeix and Ferland," Journal of Global Optimization, Springer, vol. 46(1), pages 31-47, January.
    2. R. P. Agarwal & M. Balaj & D. O’Regan, 2014. "A Common Fixed Point Theorem with Applications," Journal of Optimization Theory and Applications, Springer, vol. 163(2), pages 482-490, November.
    3. E. Allevi & A. Gnudi & I. Konnov, 2006. "The Proximal Point Method for Nonmonotone Variational Inequalities," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 63(3), pages 553-565, July.
    4. A. Daniilidis & Y. Garcia Ramos, 2007. "Some Remarks on the Class of Continuous (Semi-) Strictly Quasiconvex Functions," Journal of Optimization Theory and Applications, Springer, vol. 133(1), pages 37-48, April.
    5. D. Aussel & J. Cotrina, 2013. "Quasimonotone Quasivariational Inequalities: Existence Results and Applications," Journal of Optimization Theory and Applications, Springer, vol. 158(3), pages 637-652, September.
    6. Ravi P. Agarwal & Mircea Balaj & Donal O’Regan, 2017. "Common Fixed Point Theorems in Topological Vector Spaces via Intersection Theorems," Journal of Optimization Theory and Applications, Springer, vol. 173(2), pages 443-458, May.
    7. Arnaldo S. Brito & J. X. Cruz Neto & Jurandir O. Lopes & P. Roberto Oliveira, 2012. "Interior Proximal Algorithm for Quasiconvex Programming Problems and Variational Inequalities with Linear Constraints," Journal of Optimization Theory and Applications, Springer, vol. 154(1), pages 217-234, July.
    8. Alireza Kabgani, 2021. "Characterization of Nonsmooth Quasiconvex Functions and their Greenberg–Pierskalla’s Subdifferentials Using Semi-Quasidifferentiability notion," Journal of Optimization Theory and Applications, Springer, vol. 189(2), pages 666-678, May.
    9. Didier Aussel & Parin Chaipunya, 2024. "Variational and Quasi-Variational Inequalities Under Local Reproducibility: Solution Concept and Applications," Journal of Optimization Theory and Applications, Springer, vol. 203(2), pages 1531-1563, November.
    10. D. Aussel & N. Hadjisavvas, 2004. "On Quasimonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 121(2), pages 445-450, May.
    11. M.J. Penttinen, 2000. "Timber Harvesting with Variable Prices and Costs," Working Papers ir00045, International Institute for Applied Systems Analysis.

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