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On Controlled Variational Inequalities Involving Convex Functionals

  • Savin TreanţăEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 991)

Abstract

In this paper, by using several variational techniques and a dual gap-type functional, we study weak sharp solutions associated with a controlled variational inequality governed by convex path-independent curvilinear integral functional. Also, under some hypotheses, we establish an equivalence between the minimum principle sufficiency property and weak sharpness for a solution set of the considered controlled variational inequality.

Keywords

Controlled variational inequality Weak sharp solution Convex path-independent curvilinear integral functional 

References

  1. 1.
    Alshahrani, M., Al-Homidan S., Ansari, Q.H.: Minimum and maximum principle sufficiency properties for nonsmooth variational inequalities. Optim. Lett. 10, 805–819 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Burke, J.V., Ferris, M.C.: Weak sharp minima in mathematical programming. SIAM J. Control Optim. 31, 1340–1359 (1993)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Clarke, F.H.: Functional Analysis, Calculus of Variations and Optimal Control. Springer, London (2013)CrossRefGoogle Scholar
  4. 4.
    Ferris, M.C., Mangasarian, O.L.: Minimum principle sufficiency. Math. Program. 57, 1–14 (1992)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hiriart-Urruty, J.-B., Lemaréchal, C.: Fundamentals of Convex Analysis. Springer, Berlin (2001)CrossRefGoogle Scholar
  6. 6.
    Liu, Y., Wu, Z.: Characterization of weakly sharp solutions of a variational inequality by its primal gap function. Optim. Lett. 10, 563–576 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Mangasarian, O.L., Meyer, R.R.: Nonlinear perturbation of linear programs. SIAM J. Control Optim. 17, 745–752 (1979)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Marcotte, P., Zhu, D.: Weak sharp solutions of variational inequalities. SIAM J. Optim. 9, 179–189 (1998)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Oveisiha, M., Zafarani, J.: Generalized Minty vector variational-like inequalities and vector optimization problems in Asplund spaces. Optim. Lett. 7, 709–721 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Patriksson, M.: A unified framework of descent algorithms for nonlinear programs and variational inequalities. Ph.D. thesis, Linköping Institute of Technology (1993)Google Scholar
  11. 11.
    Polyak, B.T.: Introduction to Optimization. Optimization Software. Publications Division, New York (1987)Google Scholar
  12. 12.
    Treanţă, S.: Multiobjective fractional variational problem on higher-order jet bundles. Commun. Math. Stat. 4, 323–340 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Treanţă, S.: Higher-order Hamilton dynamics and Hamilton-Jacobi divergence PDE. Comput. Math. Appl. 75, 547–560 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Treanţă, S., Arana-Jiménez, M.: On generalized KT-pseudoinvex control problems involving multiple integral functionals. Eur. J. Control 43, 39–45 (2018)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Wu, Z., Wu, S.Y.: Weak sharp solutions of variational inequalities in Hilbert spaces. SIAM J. Optim. 14, 1011–1027 (2004)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Zhu, S.K.: Weak sharp efficiency in multiobjective optimization. Optim. Lett. 10, 1287–1301 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity Politehnica of BucharestBucharestRomania

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