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Efficiency through variational-like inequalities with Lipschitz functions

Author

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  • Gutiérrez, C.
  • Jiménez, B.
  • Novo, V.
  • Ruiz-Garzón, G.

Abstract

In this work, first we introduce several notions of invexity and pseudoinvexity for a locally Lipschitz function by means of the generalized Jacobian. We study relationships between these concepts, in particular the implications between preinvexity and invexity. Next, we obtain necessary and sufficient optimality conditions for efficient and weak efficient solutions of finite-dimensional (non necessarily Pareto) vector optimization problems with locally Lipschitz objective functions through solutions of vector variational-like inequality problems. These conditions are stated via the generalized Jacobian and under pseudoinvexity hypotheses, and they show that a vector optimization problem can be reformulated as a vector variational-like inequality problem. This work extends and improves several previous papers, where the objective function of the vector optimization problem is assumed to be differentiable, or being locally Lipschitz, the authors consider the componentwise subdifferential based on the Clarke’s generalized gradients of the components of the objective function. Throughout the paper some simple examples are given in order to illustrate the main concepts and results.

Suggested Citation

  • Gutiérrez, C. & Jiménez, B. & Novo, V. & Ruiz-Garzón, G., 2015. "Efficiency through variational-like inequalities with Lipschitz functions," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 438-449.
    • Handle: RePEc:eee:apmaco:v:259:y:2015:i:c:p:438-449
    DOI: 10.1016/j.amc.2015.02.074
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    References listed on IDEAS

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    1. Qamrul Ansari & Mahboubeh Rezaie & Jafar Zafarani, 2012. "Generalized vector variational-like inequalities and vector optimization," Journal of Global Optimization, Springer, vol. 53(2), pages 271-284, June.
    2. 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.
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    Cited by:

    1. Manuel Arana-Jiménez & Riccardo Cambini & Laura Carosi, 2018. "A reduced formulation for pseudoinvex vector functions," Annals of Operations Research, Springer, vol. 269(1), pages 21-27, October.
    2. S. K. Suneja & Sunila Sharma & Priyanka Yadav, 2018. "Generalized higher-order cone-convex functions and higher-order duality in vector optimization," Annals of Operations Research, Springer, vol. 269(1), pages 709-725, October.

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