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On Approximately Star-Shaped Functions and Approximate Vector Variational Inequalities

Author

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  • S. K. Mishra

    (Banaras Hindu University)

  • Vivek Laha

    (Banaras Hindu University)

Abstract

In this paper, we consider a vector optimization problem involving approximately star-shaped functions. We formulate approximate vector variational inequalities in terms of Fréchet subdifferentials and solve the vector optimization problem. Under the assumptions of approximately straight functions, we establish necessary and sufficient conditions for a solution of approximate vector variational inequality to be an approximate efficient solution of the vector optimization problem. We also consider the corresponding weak versions of the approximate vector variational inequalities and establish various results for approximate weak efficient solutions.

Suggested Citation

  • S. K. Mishra & Vivek Laha, 2013. "On Approximately Star-Shaped Functions and Approximate Vector Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 156(2), pages 278-293, February.
    • Handle: RePEc:spr:joptap:v:156:y:2013:i:2:d:10.1007_s10957-012-0124-4
    DOI: 10.1007/s10957-012-0124-4
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    References listed on IDEAS

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    1. Alejandro Jofré & R. Terry Rockafellar & Roger J-B. Wets, 2007. "Variational Inequalities and Economic Equilibrium," Mathematics of Operations Research, INFORMS, vol. 32(1), pages 32-50, February.
    2. Shashi Kant Mishra & Shouyang Wang & Kin Keung Lai, 2008. "V-Invex Functions and Vector Optimization," Springer Optimization and Its Applications, Springer, number 978-0-387-75446-8, June.
    3. Radu Boţ & Delia-Maria Nechita, 2011. "On the Dini-Hadamard subdifferential of the difference of two functions," Journal of Global Optimization, Springer, vol. 50(3), pages 485-502, July.
    4. S. Al-Homidan & Q. H. Ansari, 2010. "Generalized Minty Vector Variational-Like Inequalities and Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 144(1), pages 1-11, January.
    5. Y.P. Fang & N.J. Huang, 2003. "Variational-Like Inequalities with Generalized Monotone Mappings in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 118(2), pages 327-338, August.
    6. Jean-Paul Penot, 2011. "The directional subdifferential of the difference of two convex functions," Journal of Global Optimization, Springer, vol. 49(3), pages 505-519, March.
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    Cited by:

    1. Jong Kyu Kim & Salahuddin, 2020. "Local Sharp Vector Variational Type Inequality and Optimization Problems," Mathematics, MDPI, vol. 8(10), pages 1-10, October.
    2. M. Golestani & H. Sadeghi & Y. Tavan, 2018. "Nonsmooth Multiobjective Problems and Generalized Vector Variational Inequalities Using Quasi-Efficiency," Journal of Optimization Theory and Applications, Springer, vol. 179(3), pages 896-916, December.
    3. Vivek Laha & Harsh Narayan Singh, 2023. "On quasidifferentiable mathematical programs with equilibrium constraints," Computational Management Science, Springer, vol. 20(1), pages 1-20, December.

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