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Some constraint qualifications for quasiconvex vector-valued systems

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  • Satoshi Suzuki
  • Daishi Kuroiwa

Abstract

In this paper, we consider minimization problems with a quasiconvex vector-valued inequality constraint. We propose two constraint qualifications, the closed cone constraint qualification for vector-valued quasiconvex programming (the VQ-CCCQ) and the basic constraint qualification for vector-valued quasiconvex programming (the VQ-BCQ). Based on previous results by Benoist et al. (Proc Am Math Soc 13:1109–1113, 2002 ), and Suzuki and Kuroiwa (J Optim Theory Appl 149:554–563, 2011 ), and (Nonlinear Anal 74:1279–1285, 2011 ), we show that the VQ-CCCQ (resp. the VQ-BCQ) is the weakest constraint qualification for Lagrangian-type strong (resp. min–max) duality. As consequences of the main results, we study semi-definite quasiconvex programming problems in our scheme, and we observe the weakest constraint qualifications for Lagrangian-type strong and min–max dualities. Finally, we summarize the characterizations of the weakest constraint qualifications for convex and quasiconvex programming. Copyright Springer Science+Business Media, LLC. 2013

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  • 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.
    • Handle: RePEc:spr:jglopt:v:55:y:2013:i:3:p:539-548
    DOI: 10.1007/s10898-011-9807-x
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    References listed on IDEAS

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    1. Satoshi Suzuki & Daishi Kuroiwa, 2011. "On Set Containment Characterization and Constraint Qualification for Quasiconvex Programming," Journal of Optimization Theory and Applications, Springer, vol. 149(3), pages 554-563, June.
    2. Jean-Paul Penot & Michel Volle, 1990. "On Quasi-Convex Duality," Mathematics of Operations Research, INFORMS, vol. 15(4), pages 597-625, November.
    3. V. Jeyakumar, 2008. "Constraint Qualifications Characterizing Lagrangian Duality in Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 136(1), pages 31-41, January.
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    1. 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.
    2. 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.
    3. Satoshi Suzuki & Daishi Kuroiwa, 2020. "Duality Theorems for Convex and Quasiconvex Set Functions," SN Operations Research Forum, Springer, vol. 1(1), pages 1-13, March.
    4. 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.
    5. Satoshi Suzuki & Daishi Kuroiwa, 2017. "Duality Theorems for Separable Convex Programming Without Qualifications," Journal of Optimization Theory and Applications, Springer, vol. 172(2), pages 669-683, February.

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