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Minty Variational Inequalities, Increase-Along-Rays Property and Optimization1

Author

Listed:
  • G. P. Crespi

    (Université de la Vallée d’Aoste)

  • I. Ginchev

    (Technical University of Varna)

  • M. Rocca

    (University of Insubria)

Abstract

Let E be a linear space, let K $$\subseteq$$ E and f:K→ℝ . We formulate in terms of the lower Dini directional derivative problem GMVI (f ′,K ), which can be considered as a generalization of MVI (f ′,K ), the Minty variational inequality of differential type. We investigate, in the case of K star-shaped (SS), the existence of a solution x * of GMVI (f ′K ) and the property of f to increase-along-rays starting at x *, f∈IAR (K,x *). We prove that the GMVI (f ′,K ) with radially l.s.c. function f has a solution x *∈ ker K if and only if f∈IAR (K,x *). Further, we prove that the solution set of the GMVI (f ′,K ) is a convex and radially closed subset of ker K. We show also that, if the GMVI (f ′,K ) has a solution x *∈K, then x * is a global minimizer of the problem min f(x), x∈K. Moreover, we observe that the set of the global minimizers of the related optimization problem, its kernel, and the solution set of the variational inequality can be different. Finally, we prove that, in the case of a quasiconvex function f, these sets coincide.

Suggested Citation

  • G. P. Crespi & I. Ginchev & M. Rocca, 2004. "Minty Variational Inequalities, Increase-Along-Rays Property and Optimization1," Journal of Optimization Theory and Applications, Springer, vol. 123(3), pages 479-496, December.
    • Handle: RePEc:spr:joptap:v:123:y:2004:i:3:d:10.1007_s10957-004-5719-y
    DOI: 10.1007/s10957-004-5719-y
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    Citations

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    Cited by:

    1. Giovanni P. Crespi & Matteo Rocca & Carola Schrage, 2015. "Variational Inequalities Characterizing Weak Minimality in Set Optimization," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 804-824, September.
    2. Giovanni P. Crespi & Carola Schrage, 2021. "Applying set optimization to weak efficiency," Annals of Operations Research, Springer, vol. 296(1), pages 779-801, January.
    3. Ren-you Zhong & Nan-jing Huang, 2010. "Stability Analysis for Minty Mixed Variational Inequality in Reflexive Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 147(3), pages 454-472, December.
    4. M. Darabi & J. Zafarani, 2015. "Tykhonov Well-Posedness for Quasi-Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 165(2), pages 458-479, May.
    5. Q. H. Ansari & G. M. Lee, 2010. "Nonsmooth Vector Optimization Problems and Minty Vector Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 145(1), pages 1-16, April.
    6. G. P. Crespi & A. Guerraggio & M. Rocca, 2007. "Well Posedness in Vector Optimization Problems and Vector Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 132(1), pages 213-226, January.
    7. M. A. Noor & K. I. Noor & E. Al-Said, 2011. "Auxiliary Principle Technique for Solving Bifunction Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 149(2), pages 441-445, May.
    8. V. I. Ivanov, 2010. "Optimization and Variational Inequalities with Pseudoconvex Functions," Journal of Optimization Theory and Applications, Springer, vol. 146(3), pages 602-616, September.

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