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Relaxed Quasimonotone Operators and Relaxed Quasiconvex Functions

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  • M. R. Bai

    (Hunan University)

  • N. Hadjisavvas

    (University of the Aegean)

Abstract

In this paper, we introduce the class of multivalued relaxed μ quasimonotone operators and establish the existence of solutions of variational inequalities for such operators. This result is compared with a recent result of Bai et al. on densely relaxed pseudomonotone operators. A similar comparison regarding an existence result of Luc on densely pseudomonotone operators is provided. Also, we introduce a broad class of functions, called relaxed quasiconvex functions, and show that they are characterized by the relaxed μ quasimonotonicity of their subdifferentials. The results strengthen a variety of other results in the literature.

Suggested Citation

  • M. R. Bai & N. Hadjisavvas, 2008. "Relaxed Quasimonotone Operators and Relaxed Quasiconvex Functions," Journal of Optimization Theory and Applications, Springer, vol. 138(3), pages 329-339, September.
    • Handle: RePEc:spr:joptap:v:138:y:2008:i:3:d:10.1007_s10957-008-9382-6
    DOI: 10.1007/s10957-008-9382-6
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    References listed on IDEAS

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    1. D. Aussel & N. Hadjisavvas, 2004. "On Quasimonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 121(2), pages 445-450, May.
    2. 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.
    3. N. El Farouq, 2004. "Convergent Algorithm Based on Progressive Regularization for Solving Pseudomonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 120(3), pages 455-485, March.
    4. 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.
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    Cited by:

    1. M. Castellani & M. Giuli, 2013. "Refinements of existence results for relaxed quasimonotone equilibrium problems," Journal of Global Optimization, Springer, vol. 57(4), pages 1213-1227, December.

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