Advertisement

Computational Optimization and Applications

, Volume 69, Issue 3, pp 801–824 | Cite as

Augmented Lagrangian and exact penalty methods for quasi-variational inequalities

  • Christian Kanzow
  • Daniel Steck
Article
  • 370 Downloads

Abstract

A variant of the classical augmented Lagrangian method was recently proposed in Kanzow (Math Program 160(1–2, Ser. A):33–63, 2016), Pang and Fukushima (Comput Manag Sci 2(1):21–56, 2005) for the solution of quasi-variational inequalities (QVIs). In this paper, we describe an improved convergence analysis to the method. In particular, we introduce a secondary QVI as a new optimality concept for quasi-variational inequalities and use this tool to prove convergence theorems for certain popular classes of QVIs under very mild assumptions. Finally, we present a modification of the augmented Lagrangian method which turns out to be an exact penalty method, and also give detailed numerical results illustrating the performance of both methods.

Keywords

Quasi-variational inequality Augmented Lagrangian method Global convergence Feasibility Exact penalty 

References

  1. 1.
    Aussel, D., Sagratella, S.: Sufficient conditions to compute any solution of a quasivariational inequality via a variational inequality. Math. Methods Oper. Res. 85(1), 3–18 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities: Applications to Free Boundary Problems. Wiley, New York (1984)zbMATHGoogle Scholar
  3. 3.
    Bensoussan, A.: Points de Nash dans le cas de fonctionnelles quadratiques et jeux différentiels linéaires à \(N\) personnes. SIAM J. Control 12, 460–499 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bensoussan, A., Goursat, M., Lions, J.-L.: Contrôle impulsionnel et inéquations quasi-variationnelles stationnaires. C. R. Acad. Sci. Paris Sér. A-B 276, A1279–A1284 (1973)zbMATHGoogle Scholar
  5. 5.
    Beremlijski, P., Haslinger, J., Kočvara, M., Outrata, J.: Shape optimization in contact problems with Coulomb friction. SIAM J. Optim. 13(2), 561–587 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Computer Science and Applied Mathematics. Academic Press, New York (1982)zbMATHGoogle Scholar
  7. 7.
    Birgin, E.G., Fernández, D., Martínez, J.M.: The boundedness of penalty parameters in an augmented Lagrangian method with constrained subproblems. Optim. Methods Softw. 27(6), 1001–1024 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Birgin, E.G., Martínez, J.M.: Practical Augmented Lagrangian Methods for Constrained Optimization, volume 10 of Fundamentals of Algorithms. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2014)Google Scholar
  9. 9.
    Bliemer, M.C., Bovy, P.H.: Quasi-variational inequality formulation of the multiclass dynamic traffic assignment problem. Transp. Res. B: Methodol. 37(6), 501–519 (2003)CrossRefGoogle Scholar
  10. 10.
    Burke, J.V., Han, S.-P.: A robust sequential quadratic programming method. Math. Program. 43(3, (Ser. A)), 277–303 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    de André, T.A., Silva, P.J.S.: Exact penalties for variational inequalities with applications to nonlinear complementarity problems. Comput. Optim. Appl. 47(3), 401–429 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    De Luca, M., Maugeri, A.: Discontinuous quasi-variational inequalities and applications to equilibrium problems. In: Nonsmooth Optimization: Methods and Applications (Erice, 1991), pp. 70–75. Gordon and Breach, Montreux (1992)Google Scholar
  13. 13.
    Di Pillo, G., Grippo, L.: A continuously differentiable exact penalty function for nonlinear programming problems with inequality constraints. SIAM J. Control Optim. 23(1), 72–84 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Di Pillo, G., Grippo, L.: Exact penalty functions in constrained optimization. SIAM J. Control Optim. 27(6), 1333–1360 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2, Ser. A), 201–213 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Facchinei, F., Kanzow, C.: Generalized Nash equilibrium problems. Ann. Oper. Res. 175, 177–211 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Facchinei, F., Kanzow, C.: Penalty methods for the solution of generalized Nash equilibrium problems. SIAM J. Optim. 20(5), 2228–2253 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Facchinei, F., Kanzow, C., Karl, S., Sagratella, S.: The semismooth Newton method for the solution of quasi-variational inequalities. Comput. Optim. Appl. 62(1), 85–109 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Facchinei, F., Kanzow, C., Sagratella, S.: QVILIB: a library of quasi-variational inequality test problems. Pac. J. Optim. 9(2), 225–250 (2013)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Facchinei, F., Kanzow, C., Sagratella, S.: Solving quasi-variational inequalities via their KKT conditions. Math. Program. 144(1), 369–412 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Facchinei, F., Lampariello, L.: Partial penalization for the solution of generalized Nash equilibrium problems. J. Glob. Optim. 50(1), 39–57 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Fernández, D., Solodov, M.V.: Local convergence of exact and inexact augmented Lagrangian methods under the second-order sufficient optimality condition. SIAM J. Optim. 22(2), 384–407 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Fischer, A.: A special Newton-type optimization method. Optimization 24(3–4), 269–284 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Fischer, A., Herrich, M., Schönefeld, K.: Generalized Nash equilibrium problems—recent advances and challenges. Pesquisa Operacional 34, 521–558 (2014). 12CrossRefGoogle Scholar
  25. 25.
    Fukushima, M.: Restricted generalized Nash equilibria and controlled penalty algorithm. Comput. Manag. Sci. 8(3), 201–218 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hammerstein, P., Selten, R.: Game theory and evolutionary biology. In: Aumann, R.J., Hart, S. (eds.) Handbook of Game Theory with Economic Applications, Vol. II, Volume 11 of Handbooks in Economics, pp. 929–993. North-Holland, Amsterdam (1994)Google Scholar
  27. 27.
    Harker, P.T.: Generalized Nash games and quasi-variational inequalities. Eur. J. Oper. Res. 54(1), 81–94 (1991)CrossRefzbMATHGoogle Scholar
  28. 28.
    Haslinger, J., Panagiotopoulos, P.D.: The reciprocal variational approach to the Signorini problem with friction. Approximation results. Proc. R. Soc. Edinb. Sect. A 98(3–4), 365–383 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Hintermüller, M., Rautenberg, C.N.: A sequential minimization technique for elliptic quasi-variational inequalities with gradient constraints. SIAM J. Optim. 22(4), 1224–1257 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Hintermüller, M., Rautenberg, C.N.: Parabolic quasi-variational inequalities with gradient-type constraints. SIAM J. Optim. 23(4), 2090–2123 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Ichiishi, T.: Game Theory for Economic Analysis. Economic Theory, Econometrics, and Mathematical Economics. Academic Press Inc, New York (1983)zbMATHGoogle Scholar
  32. 32.
    Jing-Yuan, W., Smeers, Y.: Spatial oligopolistic electricity models with cournot generators and regulated transmission prices. Oper. Res. 47(1), 102–112 (1999)CrossRefzbMATHGoogle Scholar
  33. 33.
    Kanzow, C.: On the multiplier-penalty-approach for quasi-variational inequalities. Math. Program. 160(1–2, Ser. A), 33–63 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Kanzow, C., Steck, D.: Augmented Lagrangian methods for the solution of generalized Nash equilibrium problems. SIAM J. Optim. 26(4), 2034–2058 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Kanzow, C., Steck, D.: An example comparing the standard and safeguarded augmented Lagrangian methods. Oper. Res. Lett. 45(6), 598–603 (2017)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Kanzow, C., Steck, D.: On Error Bounds and Multiplier Methods for Variational Problems in Banach Spaces. Technical Report, Institute of Mathematics, University of Würzburg (2017)Google Scholar
  37. 37.
    Kharroubi, I., Ma, J., Pham, H., Zhang, J.: Backward SDEs with constrained jumps and quasi-variational inequalities. Ann. Probab. 38(2), 794–840 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Kravchuk, A.S., Neittaanmäki, P.J.: Variational and Quasi-variational Inequalities in Mechanics, volume of 147 Solid Mechanics and Its Applications. Springer, Dordrecht (2007)CrossRefzbMATHGoogle Scholar
  39. 39.
    Kunze, M., Rodrigues, J.F.: An elliptic quasi-variational inequality with gradient constraints and some of its applications. Math. Methods Appl. Sci. 23(10), 897–908 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Lucidi, S.: New results on a continuously differentiable exact penalty function. SIAM J. Optim. 2(4), 558–574 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Mosco, U.: Implicit variational problems and quasi variational inequalities. In: Nonlinear Operators and the Calculus of Variations (Summer School, Univ. Libre Bruxelles, Brussels, 1975), pp. 83–156. Lecture Notes in Mathematics, vol. 543. Springer, Berlin (1976)Google Scholar
  42. 42.
    Outrata, J., Kočvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, volume 28 of Nonconvex Optimization and Its Applications. Theory, Applications and Numerical Results. Kluwer, Dordrecht (1998)Google Scholar
  43. 43.
    Outrata, J.V., Kočvara, M.: On a class of quasi-variational inequalities. Optim. Methods Softw. 5(4), 275–295 (1995)CrossRefzbMATHGoogle Scholar
  44. 44.
    Pang, J.-S., Fukushima, M.: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Comput. Manag. Sci. 2(1), 21–56 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Scrimali, L.: Quasi-variational inequalities in transportation networks. Math. Models Methods Appl. Sci. 14(10), 1541–1560 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of WürzburgWürzburgGermany

Personalised recommendations