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Characterizations of the solution set for quasiconvex programming in terms of Greenberg–Pierskalla subdifferential

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  • Satoshi Suzuki
  • Daishi Kuroiwa

Abstract

In convex programming, characterizations of the solution set in terms of the subdifferential have been investigated by Mangasarian. An invariance property of the subdifferential of the objective function is studied, and as a consequence, characterizations of the solution set by any solution point and any point in the relative interior of the solution set are given. In quasiconvex programming, however, characterizations of the solution set by any solution point and an invariance property of Greenberg–Pierskalla subdifferential, which is one of the well known subdifferential for quasiconvex functions, have not been studied yet as far as we know. In this paper, we study characterizations of the solution set for quasiconvex programming in terms of Greenberg–Pierskalla subdifferential. To the purpose, we show an invariance property of Greenberg–Pierskalla subdifferential, and we introduce a necessary and sufficient optimality condition by Greenberg–Pierskalla subdifferential. Also, we compare our results with previous ones. Especially, we prove some of Mangasarian’s characterizations as corollaries of our results. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:62:y:2015:i:3:p:431-441
    DOI: 10.1007/s10898-014-0255-2
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    References listed on IDEAS

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    1. J.P. Penot, 2003. "Characterization of Solution Sets of Quasiconvex Programs," Journal of Optimization Theory and Applications, Springer, vol. 117(3), pages 627-636, June.
    2. Satoshi Suzuki & Daishi Kuroiwa, 2013. "Some constraint qualifications for quasiconvex vector-valued systems," Journal of Global Optimization, Springer, vol. 55(3), pages 539-548, March.
    3. V. Jeyakumar & G. M. Lee & N. Dinh, 2004. "Lagrange Multiplier Conditions Characterizing the Optimal Solution Sets of Cone-Constrained Convex Programs," Journal of Optimization Theory and Applications, Springer, vol. 123(1), pages 83-103, October.
    4. X. M. Yang, 2009. "On Characterizing the Solution Sets of Pseudoinvex Extremum Problems," Journal of Optimization Theory and Applications, Springer, vol. 140(3), pages 537-542, March.
    5. Jean-Paul Penot & Michel Volle, 1990. "On Quasi-Convex Duality," Mathematics of Operations Research, INFORMS, vol. 15(4), pages 597-625, November.
    6. Z. L. Wu & S. Y. Wu, 2006. "Characterizations of the Solution Sets of Convex Programs and Variational Inequality Problems," Journal of Optimization Theory and Applications, Springer, vol. 130(2), pages 341-360, August.
    7. Vsevolod Ivanov, 2013. "Characterizations of pseudoconvex functions and semistrictly quasiconvex ones," Journal of Global Optimization, Springer, vol. 57(3), pages 677-693, November.
    8. Satoshi Suzuki & Daishi Kuroiwa, 2012. "Necessary and Sufficient Constraint Qualification for Surrogate Duality," Journal of Optimization Theory and Applications, Springer, vol. 152(2), pages 366-377, February.
    9. 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.
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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. 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.
    3. Satoshi Suzuki, 2021. "Karush–Kuhn–Tucker type optimality condition for quasiconvex programming in terms of Greenberg–Pierskalla subdifferential," Journal of Global Optimization, Springer, vol. 79(1), pages 191-202, January.
    4. Xiangkai Sun & Kok Lay Teo & Liping Tang, 2019. "Dual Approaches to Characterize Robust Optimal Solution Sets for a Class of Uncertain Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 182(3), pages 984-1000, September.
    5. Shashi Kant Mishra & Vivek Laha & Mohd Hassan, 2024. "On Quasiconvex Multiobjective Optimization and Variational Inequalities Using Greenberg–Pierskalla Based Generalized Subdifferentials," Journal of Optimization Theory and Applications, Springer, vol. 202(3), pages 1169-1186, September.
    6. Nader Kanzi & Majid Soleimani-damaneh, 2020. "Characterization of the weakly efficient solutions in nonsmooth quasiconvex multiobjective optimization," Journal of Global Optimization, Springer, vol. 77(3), pages 627-641, July.
    7. 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.

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