Journal of Global Optimization

, Volume 70, Issue 2, pp 401–411 | Cite as

Inverse problems for quasi-variational inequalities

  • Akhtar A. Khan
  • Dumitru Motreanu


In this short note, our aim is to investigate the inverse problem of parameter identification in quasi-variational inequalities. We develop an abstract nonsmooth regularization approach that subsumes the total variation regularization and permits the identification of discontinuous parameters. We study the inverse problem in an optimization setting using the output-least squares formulation. We prove the existence of a global minimizer and give convergence results for the considered optimization problem. We also discretize the identification problem for quasi-variational inequalities and provide the convergence analysis for the discrete problem. We give an application to the gradient obstacle problem.


Inverse problems Regularization Output least-squares Quasi-variational inequalities 

Mathematics Subject Classification

35R30 49N45 65J20 65J22 65M30 



The authors are grateful to the reviewers for the valuable remarks. Akhtar Khan is supported by National Science Foundation Grant 005613-002 and RIT’s COS FEAD Grant for 2016–2017.


  1. 1.
    Acar, R., Vogel, C.R.: Analysis of bounded variation penalty methods for ill-posed problems. Inverse Probl. 10, 1217–1229 (1994)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Barbagallo, A., Mauro, P.: A general quasi-variational problem of Cournot–Nash type and its inverse formulation. J. Optim. Theory Appl. 170(2), 476–492 (2016)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Barrett, J.W., Prigozhin, L.: A quasi-variational inequality problem arising in the modeling of growing sandpiles. ESAIM Math. Model. Numer. Anal. 47, 1133–1165 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bensoussan, A., Lions, J.-L.: Optimal impulse and continuous control: method of nonlinear quasivariational inequalities. Trudy Mat. Inst. Steklov. 134, 5–22 (1975)MathSciNetMATHGoogle Scholar
  5. 5.
    Boiger, R., Kaltenbacher, B.: An online parameter identification method for time dependent partial differential equations. Inverse Probl. 32(4), 045006 (2016)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Ciarciá, C., Daniele, P.: New existence theorems for quasi-variational inequalities and applications to financial models. Eur. J. Oper. Res. 251, 288–299 (2016)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Clason, C.: \(L^{\infty }\) fitting for inverse problems with uniform noise. Inverse Probl. 28(10), 104007 (2012)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Giannessi, F., Khan, A.A.: Regularization of non-coercive quasi variational inequalities. Control Cybern. 29, 91–110 (2000)MathSciNetMATHGoogle Scholar
  9. 9.
    Gockenbach, M.S., Khan, A.A.: An abstract framework for elliptic inverse problems. I. An output least-squares approach. Math. Mech. Solids 12, 259–276 (2007)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hintermüller, M.: Inverse coefficient problems for variational inequalities: optimality conditions and numerical realization. M2AN Math. Model. Numer. Anal. 35, 129–152 (2001)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Jadamba, B., Khan, A.A., Rus, G., Sama, M., Winkler, B.: A new convex inversion framework for parameter identification in saddle point problems with an application to the elasticity imaging inverse problem of predicting tumor location. SIAM J. Appl. Math. 74, 1486–1510 (2014)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kano, R., Kenmochi, N., Murase, Y.: Existence theorems for elliptic quasi-variational inequalities in Banach spaces. In: Chipot, M., Lin, C.-S., Tsai, D.-H. (eds.) Recent Advances in Nonlinear Analysis, pp. 149–169. World Scientific Publishing, Hackensack (2008)CrossRefGoogle Scholar
  13. 13.
    Khan, A.A., Motreanu, D.: Existence theorems for elliptic and evolutionary variational and quasi-variational inequalities. J. Optim. Theory Appl. 167(3), 1136–1161 (2015)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Khan, A.A., Tammer, C., Zalinescu, C.: Regularization of quasi-variational inequalities. Optimization 64, 1703–1724 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kluge, R. (ed.): On some parameter determination problems and quasi-variational inequalities. In: Theory of Nonlinear Operators, vol. 6, pp. 129–139. Akademie-Verlag, Berlin (1978).Google Scholar
  16. 16.
    Lenzen, F., Becker, F., Lellmann, J., Petra, S., Schnörr, C.: A class of quasi-variational inequalities for adaptive image denoising and decomposition. Comput. Optim. Appl. 54(2), 371–398 (2013)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Liu, F., Nashed, M.Z.: Regularization of nonlinear ill-posed variational inequalities and convergence rates. Set Valued Anal. 6, 313–344 (1998)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Mosco, U.: Implicit variational problems and quasi variational inequalities. In: Nonlinear Operators and the Calculus of Variations, Lecture Notes in Mathematics, pp. 83–156. Springer, Berlin (1976)Google Scholar
  19. 19.
    Motreanu, D., Sofonea, M.: Quasivariational inequalities and applications in frictional contact problems with normal compliance. Adv. Math. Sci. Appl. 10, 103–118 (2000)MathSciNetMATHGoogle Scholar
  20. 20.
    Nashed, M.Z., Scherzer, O.: Least squares and bounded variation regularization with nondifferentiable functionals. Numer. Funct. Anal. Optim. 19, 873–901 (1998)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Resmerita, E.: Regularization of ill-posed problems in Banach spaces: convergence rates. Inverse Probl. 21, 1303–1314 (2005)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Scrimali, L.: Evolutionary quasi-variational inequalities and the dynamic multiclass network equilibrium problem. Numer. Funct. Anal. Optim. 35(7–9), 1225–1244 (2014)MathSciNetCrossRefMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Center for Applied and Computational Mathematics, School of Mathematical SciencesRochester Institute of TechnologyRochesterUSA
  2. 2.Département de MathématiquesUniversité de PerpignanPerpignanFrance

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