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Higher-order Pseudoconvex Functions

In: Generalized Convexity and Related Topics

Author

Listed:
  • Ivan Ginchev

    (Technical University of Varna)

  • Vsevolod I. Ivanov

    (Technical University of Varna)

Abstract

Summary In terms of n-th order Dini directional derivative with n positive integer we define n-pseudoconvex functions being a generalization of the usual pseudoconvex functions. Again with the n-th order Dini derivative we define n-stationary points, and prove that a point x 0 is a global minimizer of a n-pseudoconvex function f if and only if x 0 is a n-stationary point of f. Our main result is the following. A radially continuous function f defined on a radially open convex set in a real linear space is n-pseudoconvex if and only if f is quasiconvex function and any n-stationary point is a global minimizer. This statement generalizes the results of Crouzeix, Ferland, Math. Program. 23 (1982), 193–205, and Komlósi, Math. Program. 26 (1983), 232–237. We study also other aspects of the n-pseudoconvex functions, for instance their relations to variational inequalities.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:lnechp:978-3-540-37007-9_14
    DOI: 10.1007/978-3-540-37007-9_14
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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. Vsevolod Ivanov, 2013. "Characterizations of pseudoconvex functions and semistrictly quasiconvex ones," Journal of Global Optimization, Springer, vol. 57(3), pages 677-693, November.
    3. Vsevolod I. Ivanov, 2013. "Optimality Conditions and Characterizations of the Solution Sets in Generalized Convex Problems and Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 158(1), pages 65-84, July.
    4. Vsevolod I. Ivanov, 2015. "Second-Order Optimality Conditions for Vector Problems with Continuously Fréchet Differentiable Data and Second-Order Constraint Qualifications," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 777-790, September.
    5. 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.

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