Skip to main content
SpringerLink
Log in

On a Smooth Dual Gap Function for a Class of Quasi-Variational Inequalities

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

A well-known technique for the solution of quasi-variational inequalities (QVIs) consists in the reformulation of this problem as a constrained or unconstrained optimization problem by means of so-called gap functions. In contrast to standard variational inequalities, however, these gap functions turn out to be nonsmooth in general. Here, it is shown that one can obtain an unconstrained optimization reformulation of a class of QVIs with affine operator by using a continuously differentiable dual gap function. This extends an idea from Dietrich (J. Math. Anal. Appl. 235:380–393 [24]). Some numerical results illustrate the practical behavior of this dual gap function approach.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Bensoussan, A., Goursat, M., Lions, J.-L.: Contrôle impulsionnel et inéquations quasi-variationnelles stationnaires. C. R. Acad. Sci. Paris A 276, 1279–1284 (1973)

    MathSciNet  MATH  Google Scholar 

  2. Bensoussan, A., Lions, J.-L.: Nouvelle formulation de problèmes de contrôle impulsionnel et applications. C. R. Acad. Sci. Paris A 276, 1189–1192 (1973)

    MathSciNet  MATH  Google Scholar 

  3. Bensoussan, A., Lions, J.-L.: Nouvelles méthodes en contrôle impulsionnel. Appl. Math. Optim. 1, 289–312 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  4. Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities: Applications to Free Boundary Problems. Wiley, New York (1984)

    MATH  Google Scholar 

  5. Facchinei, F., Kanzow, C., Sagratella, S.: QVILIB: A library of quasi-variational inequality test problems. Pac. J. Optim. 9, 225–250 (2013)

    MathSciNet  MATH  Google Scholar 

  6. Chan, D., Pang, J.-S.: The generalized quasi-variational inequality problem. Math. Oper. Res. 7, 211–222 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  7. Nesterov, Y., Scrimali, L.: Solving strongly monotone variational and quasi-variational inequalities. CORE Discussion Paper 2006/107, Catholic University of Louvain, Center for Operations Research and Econometrics (2006)

  8. Noor, M.A.: An iterative scheme for a class of quasi variational inequalities. J. Math. Anal. Appl. 110, 463–468 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  9. Noor, M.A.: On general quasi-variational inequalities. J. King Saud Univ. 24, 81–88 (2012)

    Article  Google Scholar 

  10. Ryazantseva, I.P.: First-order methods for certain quasi-variational inequalities in Hilbert space. Comput. Math. Math. Phys. 47, 183–190 (2007)

    Article  MathSciNet  Google Scholar 

  11. Siddiqi, A.H., Ansari, Q.H.: Strongly nonlinear quasivariational inequalities. J. Math. Anal. Appl. 149, 444–450 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  12. Facchinei, F., Kanzow, C., Sagratella, S.: Solving quasi-variational inequalities via their KKT conditions. Math. Program. doi:10.1007/s10107-013-0637-0

  13. Pang, J.-S., Fukushima, M.: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Comput. Manag. Sci. 2, 21–56 (2005). Erratum: ibid 6, 373–375 (2009)

    Google Scholar 

  14. Outrata, J.V., Kocvara, M.: On a class of quasi-variational inequalities. Optim. Methods Softw. 5, 275–295 (1995)

    Article  Google Scholar 

  15. Outrata, J.V., Kocvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory. Applications and Numerical Results. Kluwer Academic Publishers, Dordrecht (1998)

    Book  MATH  Google Scholar 

  16. Outrata, J.V., Zowe, J.: A Newton method for a class of quasi-variational inequalities. Comput. Optim. Appl. 4, 5–21 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. Aussel, D., Correa, R., Marechal, M.: Gap functions for quasivariational inequalities and generalized Nash equilibrium problems. J. Optim. Theory Appl. 151, 474–488 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dietrich, H.: Optimal control problems for certain quasivariational inequalities. Optimization 49, 67–93 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Fukushima, M.: A class of gap functions for quasi-variational inequality problems. J. Ind. Manag. Optim. 3, 165–171 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Giannessi, F.: Separation of sets and gap functions for quasi-variational inequalities. In: Giannessi, F., Maugeri, A. (eds.) Variational Inequality and Network Equilibrium Problems, pp. 101–121. Plenum Press, New York (1995)

    Chapter  Google Scholar 

  21. Harms, N., Kanzow, C., Stein, O.: Smoothness properties of a regularized gap function for quasi-variational inequalities. Optimization. doi:10.1080/10556788.2013.841694

  22. Taji, K.: On gap functions for quasi-variational inequalities. Abstr. Appl. Anal., Article ID 531361 (2008)

  23. Fukushima, M.: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Program. 53, 99–110 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  24. Dietrich, H.: A smooth dual gap function solution to a class of quasivariational inequalities. J. Math. Anal. Appl. 235, 380–393 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  25. Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. A Series of Comprehensive Studies in Mathematics, vol. 317. Springer, Berlin (1998)

    Google Scholar 

  26. Hiriart-Urruty, J.B., Lemaréchal, C.: Fundamentals of Convex Analysis. Grundlehren Text Editions. Springer, Berlin (2001)

    Book  Google Scholar 

  27. Horst, T., Thoai, N.V.: DC programming: overview. J. Optim. Theory Appl. 103, 1–43 (1999)

    Article  MathSciNet  Google Scholar 

  28. Toland, J.F.: Duality in non-convex optimization. J. Math. Anal. Appl. 66, 399–415 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  29. Singer, I.: A Fenchel-Rockafellar type duality theorem for maximization. Bull. Aust. Math. Soc. 20, 193–198 (1979)

    Article  MATH  Google Scholar 

  30. Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)

    MATH  Google Scholar 

  31. Hogan, W.W.: Point-to-set maps in mathematical programming. SIAM Rev. 15, 591–603 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  32. Toland, J.F.: A duality principle for non-convex optimisation and the calculus of variations. Arch. Ration. Mech. Anal. 71, 41–61 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  33. Janin, R.: Directional derivative of the marginal function in nonlinear programming. Math. Program. Stud. 21, 110–126 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  34. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vols. I and II. Springer, New York (2003)

    Google Scholar 

  35. Qi, L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18, 227–244 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  36. Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  37. Raydan, M.: The Barzilai and Borwein gradient method for the large unconstrained minimization problem. SIAM J. Optim. 7, 26–33 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  38. Moré, J.J., Sorensen, D.C.: Newton’s method. In: Golub, G.H. (ed.) Studies in Numerical Analysis, pp. 29–82. The Mathematical Association of America, Washington, DC (1984)

    Google Scholar 

  39. Sun, D., Fukushima, M., Qi, L.: A computable generalized Hessian of the D-gap function and Newton-type methods for variational inequality problems. In: Ferris, M.C., Pang, J.S. (eds.) Complementarity and Variational Problems: State of the Art, pp. 452–472. SIAM, Philadelphia (1997)

    Google Scholar 

  40. von Heusinger, A., Kanzow, C., Fukushima, M.: Newton’s method for computing a normalized equilibrium in the generalized Nash game through fixed point formulation. Math. Program. 132, 99–123 (2012)

    Google Scholar 

  41. Noor, M.A., Noor, K.I., Khan, A.G.: Some iterative schemes for solving extended general quasi variational inequalities. Appl. Math. Inform. Sci. 7, 917–925 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank Oliver Stein for his comments on an earlier draft of this paper. This research was partially supported by the Deutsche Forschungsgemeinschaft (DFG) under Grant KA 1296/18-1 as well as by a Grant from the international doctorate program “Identification, Optimization, and Control with Applications in Modern Technologies” within the Elite-Network of Bavaria.

Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Würzburg, Emil-Fischer-Str. 30, 97074 , Würzburg, Germany

    Nadja Harms, Tim Hoheisel & Christian Kanzow

Authors
  1. Nadja Harms
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tim Hoheisel
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Christian Kanzow
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christian Kanzow.

Additional information

Communicated by Patrice Marcotte.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Harms, N., Hoheisel, T. & Kanzow, C. On a Smooth Dual Gap Function for a Class of Quasi-Variational Inequalities. J Optim Theory Appl 163, 413–438 (2014). https://doi.org/10.1007/s10957-014-0536-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-014-0536-4

Keywords

Mathematics Subject Classification (2000)

Search

Navigation