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Maximizing and minimizing quasiconvex functions: related properties, existence and optimality conditions via radial epiderivatives

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  • Fabián Flores-Bazán
  • Fernando Flores-Bazán
  • Cristián Vera

Abstract

This paper deals with maximization and minimization of quasiconvex functions in a finite dimensional setting. Firstly, some existence results on closed convex sets, possibly containing lines, are presented. This is given via a careful study of reduction to the boundary and/or extremality of the feasible set. Necessary or sufficient optimality conditions are derived in terms of radial epiderivatives. Then, the problem of minimizing quasiconvex functions are analyzed via asymptotic analysis. Finally, some attempts to define asymptotic functions under quasiconvexity are also outlined. Several examples illustrating the applicability of our results are shown. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • Fabián Flores-Bazán & Fernando Flores-Bazán & Cristián Vera, 2015. "Maximizing and minimizing quasiconvex functions: related properties, existence and optimality conditions via radial epiderivatives," Journal of Global Optimization, Springer, vol. 63(1), pages 99-123, September.
    • Handle: RePEc:spr:jglopt:v:63:y:2015:i:1:p:99-123
    DOI: 10.1007/s10898-015-0267-6
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    Citations

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    Cited by:

    1. Alfredo Iusem & Felipe Lara, 2019. "Optimality Conditions for Vector Equilibrium Problems with Applications," Journal of Optimization Theory and Applications, Springer, vol. 180(1), pages 187-206, January.
    2. A. Iusem & F. Lara, 2022. "Proximal Point Algorithms for Quasiconvex Pseudomonotone Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 443-461, June.
    3. Nicolas Hadjisavvas & Felipe Lara & Dinh The Luc, 2020. "A general asymptotic function with applications in nonconvex optimization," Journal of Global Optimization, Springer, vol. 78(1), pages 49-68, September.
    4. Nicolas Hadjisavvas & Felipe Lara & Juan Enrique Martínez-Legaz, 2019. "A Quasiconvex Asymptotic Function with Applications in Optimization," Journal of Optimization Theory and Applications, Springer, vol. 180(1), pages 170-186, January.
    5. Fabián Flores-Bazán & William Echegaray & Fernando Flores-Bazán & Eladio Ocaña, 2017. "Primal or dual strong-duality in nonconvex optimization and a class of quasiconvex problems having zero duality gap," Journal of Global Optimization, Springer, vol. 69(4), pages 823-845, December.
    6. F. Flores-Bazán & N. Hadjisavvas & F. Lara & I. Montenegro, 2016. "First- and Second-Order Asymptotic Analysis with Applications in Quasiconvex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 170(2), pages 372-393, August.
    7. Fabián Flores-Bazán & Filip Thiele, 2022. "On the Lower Semicontinuity of the Value Function and Existence of Solutions in Quasiconvex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 195(2), pages 390-417, November.
    8. Felipe Lara, 2020. "Optimality Conditions for Nonconvex Nonsmooth Optimization via Global Derivatives," Journal of Optimization Theory and Applications, Springer, vol. 185(1), pages 134-150, April.
    9. Felipe Lara & Rubén López, 2017. "Formulas for Asymptotic Functions via Conjugates, Directional Derivatives and Subdifferentials," Journal of Optimization Theory and Applications, Springer, vol. 173(3), pages 793-811, June.
    10. Alfredo Iusem & Felipe Lara, 2019. "Existence Results for Noncoercive Mixed Variational Inequalities in Finite Dimensional Spaces," Journal of Optimization Theory and Applications, Springer, vol. 183(1), pages 122-138, October.

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