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Second order necessary conditions in set constrained differentiable vector optimization

Author

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  • Bienvenido Jiménez
  • Vicente Novo

Abstract

We state second order necessary optimality conditions for a vector optimization problem with an arbitrary feasible set and an order in the final space given by a pointed convex cone with nonempty interior. We establish, in finite-dimensional spaces, second order optimality conditions in dual form by means of Lagrange multipliers rules when the feasible set is defined by a function constrained to a set with convex tangent cone. To pass from general conditions to Lagrange multipliers rules, a generalized Motzkin alternative theorem is provided. All the involved functions are assumed to be twice Fréchet differentiable. Copyright Springer-Verlag 2003

Suggested Citation

  • Bienvenido Jiménez & Vicente Novo, 2003. "Second order necessary conditions in set constrained differentiable vector optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 58(2), pages 299-317, November.
    • Handle: RePEc:spr:mathme:v:58:y:2003:i:2:p:299-317
    DOI: 10.1007/s001860300283
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    Citations

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    Cited by:

    1. S. J. Li & K. L. Teo & X. Q. Yang, 2008. "Higher-Order Optimality Conditions for Set-Valued Optimization," Journal of Optimization Theory and Applications, Springer, vol. 137(3), pages 533-553, June.
    2. Giorgio, 2019. "On Second-Order Optimality Conditions in Smooth Nonlinear Programming Problems," DEM Working Papers Series 171, University of Pavia, Department of Economics and Management.
    3. Ram U. Verma & G. J. Zalmai, 2018. "Parameter-free duality models and applications to semiinfinite minmax fractional programming based on second-order ( $$\phi ,\eta ,\rho ,\theta ,{\tilde{m}}$$ ϕ , η , ρ , θ , m ~ )-sonvexities," OPSEARCH, Springer;Operational Research Society of India, vol. 55(2), pages 381-410, June.
    4. P. Q. Khanh & N. D. Tuan, 2008. "Higher-Order Variational Sets and Higher-Order Optimality Conditions for Proper Efficiency in Set-Valued Nonsmooth Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 139(2), pages 243-261, November.
    5. Anulekha Dhara & Aparna Mehra, 2013. "Second-Order Optimality Conditions in Minimax Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 156(3), pages 567-590, March.
    6. María C. Maciel & Sandra A. Santos & Graciela N. Sottosanto, 2011. "On Second-Order Optimality Conditions for Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 149(2), pages 332-351, May.
    7. J. Jahn & A. A. Khan & P. Zeilinger, 2005. "Second-Order Optimality Conditions in Set Optimization," Journal of Optimization Theory and Applications, Springer, vol. 125(2), pages 331-347, May.
    8. Tuan, Nguyen Dinh, 2015. "First and second-order optimality conditions for nonsmooth vector optimization using set-valued directional derivatives," Applied Mathematics and Computation, Elsevier, vol. 251(C), pages 300-317.
    9. Bienvenido Jiménez & Vicente Novo, 2008. "Higher-order optimality conditions for strict local minima," Annals of Operations Research, Springer, vol. 157(1), pages 183-192, January.
    10. P. Q. Khanh & N. D. Tuan, 2008. "Variational Sets of Multivalued Mappings and a Unified Study of Optimality Conditions," Journal of Optimization Theory and Applications, Springer, vol. 139(1), pages 47-65, October.
    11. C. Gutiérrez & B. Jiménez & V. Novo, 2006. "On Approximate Efficiency in Multiobjective Programming," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 64(1), pages 165-185, August.
    12. Giorgio Giorgi, 2019. "Notes on Constraint Qualifications for Second-Order Optimality Conditions," Journal of Mathematics Research, Canadian Center of Science and Education, vol. 11(5), pages 16-32, October.
    13. C. Gutiérrez & B. Jiménez & V. Novo, 2009. "New Second-Order Directional Derivative and Optimality Conditions in Scalar and Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 142(1), pages 85-106, July.
    14. Nguyen Thi Toan & Le Quang Thuy & Nguyen Tuyen & Yi-Bin Xiao, 2021. "Second-order KKT optimality conditions for multiobjective discrete optimal control problems," Journal of Global Optimization, Springer, vol. 79(1), pages 203-231, January.
    15. Do Van Luu, 2018. "Second-order necessary efficiency conditions for nonsmooth vector equilibrium problems," Journal of Global Optimization, Springer, vol. 70(2), pages 437-453, February.
    16. Bui Trong Kien & Trinh Duy Binh, 2023. "On the second-order optimality conditions for multi-objective optimal control problems with mixed pointwise constraints," Journal of Global Optimization, Springer, vol. 85(1), pages 155-183, January.

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