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Characterizations of pseudoconvex functions and semistrictly quasiconvex ones

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  • Vsevolod Ivanov

Abstract

In this paper, we provide some new necessary and sufficient conditions for pseudoconvexity and semistrict quasiconvexity of a given proper extended real-valued function in terms of the Clarke–Rockafellar subdifferential. Further, we extend to programs with pseudoconvex objective function two earlier characterizations of the solutions set of a set constrained nonlinear programming problem due to Mangasarian (Oper Res Lett 7:21–26, 1988 ). A positive function p appears in the most results. It is replaced by the number 1 if the function is convex and its domain of definition is convex, too. Copyright Springer Science+Business Media, LLC. 2013

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  • Vsevolod Ivanov, 2013. "Characterizations of pseudoconvex functions and semistrictly quasiconvex ones," Journal of Global Optimization, Springer, vol. 57(3), pages 677-693, November.
    • Handle: RePEc:spr:jglopt:v:57:y:2013:i:3:p:677-693
    DOI: 10.1007/s10898-012-9956-6
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    1. Diewert, W. E. & Avriel, M. & Zang, I., 1981. "Nine kinds of quasiconcavity and concavity," Journal of Economic Theory, Elsevier, vol. 25(3), pages 397-420, December.
    2. V. I. Ivanov, 2010. "Optimization and Variational Inequalities with Pseudoconvex Functions," Journal of Optimization Theory and Applications, Springer, vol. 146(3), pages 602-616, September.
    3. 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.
    4. Ivan Ginchev & Vsevolod I. Ivanov, 2007. "Higher-order Pseudoconvex Functions," Lecture Notes in Economics and Mathematical Systems, in: Generalized Convexity and Related Topics, pages 247-264, Springer.
    5. 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.
    6. Alberto Cambini & Laura Martein, 2009. "Generalized Convexity and Optimization," Lecture Notes in Economics and Mathematical Systems, Springer, number 978-3-540-70876-6, February.
    7. X. F. Li & J. L. Dong & Q. H. Liu, 1997. "Lipschitz B-Vex Functions and Nonsmooth Programming," Journal of Optimization Theory and Applications, Springer, vol. 93(3), pages 557-574, June.
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    Cited by:

    1. Vsevolod I. Ivanov, 2020. "Characterization of Radially Lower Semicontinuous Pseudoconvex Functions," Journal of Optimization Theory and Applications, Springer, vol. 184(2), pages 368-383, February.
    2. Mohammad Taghi Nadi & Jafar Zafarani, 2023. "Second-order characterization of convex mappings in Banach spaces and its applications," Journal of Global Optimization, Springer, vol. 86(4), pages 1005-1023, August.
    3. Satoshi Suzuki, 2019. "Optimality Conditions and Constraint Qualifications for Quasiconvex Programming," Journal of Optimization Theory and Applications, Springer, vol. 183(3), pages 963-976, December.
    4. Vsevolod I. Ivanov, 2019. "Characterizations of Solution Sets of Differentiable Quasiconvex Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 181(1), pages 144-162, April.
    5. Satoshi Suzuki & Daishi Kuroiwa, 2015. "Characterizations of the solution set for quasiconvex programming in terms of Greenberg–Pierskalla subdifferential," Journal of Global Optimization, Springer, vol. 62(3), pages 431-441, July.

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