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First and second-order optimality conditions for nonsmooth vector optimization using set-valued directional derivatives

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  • Tuan, Nguyen Dinh

Abstract

We investigate a nonsmooth vector optimization problem with a feasible set defined by a generalized inequality constraint, an equality constraint and a set constraint. Both necessary and sufficient optimality conditions of first and second-order for weak solutions and firm solutions are established in terms of Fritz-John–Lagrange multiplier rules using set-valued directional derivatives and tangent cones and second-order tangent sets. We impose steadiness and strict differentiability for first and second-order necessary conditions, respectively; stability and l-stability for first and second-order sufficient conditions, respectively. The obtained results improve or include some recent known ones. Several illustrative examples are also provided.

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  • 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.
    • Handle: RePEc:eee:apmaco:v:251:y:2015:i:c:p:300-317
    DOI: 10.1016/j.amc.2014.11.061
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    1. V. Jeyakumar & D.T. LUC, 2008. "Nonsmooth Vector Functions and Continuous Optimization," Springer Optimization and Its Applications, Springer, number 978-0-387-73717-1, September.
    2. P. Q. Khanh & N. D. Tuan, 2007. "Optimality Conditions for Nonsmooth Multiobjective Optimization Using Hadamard Directional Derivatives," Journal of Optimization Theory and Applications, Springer, vol. 133(3), pages 341-357, June.
    3. 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.
    4. Yukihiro Maruyama, 1990. "Second-Order Necessary Conditions for Nonlinear Optimization Problems in Banach Spaces and Their Application to an Optimal Control Problem," Mathematics of Operations Research, INFORMS, vol. 15(3), pages 467-482, August.
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    Cited by:

    1. Thinh, Vo Duc & Chuong, Thai Doan & Le Hoang Anh, Nguyen, 2023. "Formulas of first-ordered and second-ordered generalization differentials for convex robust systems with applications," Applied Mathematics and Computation, Elsevier, vol. 455(C).

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