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Right-Hand Side Decomposition for Variational Inequalities

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Abstract

We consider a general class of variational inequality problems in a finite-dimensional space setting. The cost mapping need not be the gradient of any function. By using a right-hand side allocation technique, we transform such a problem into a collection of small-dimensional variational inequalities. The master problem is a set-valued variational inequality. We suggest a general iterative method for the problem obtained, which is convergent under monotonicity assumptions. We also show that regularization of partial problems enables us to create single-valued approximations for the cost mapping of the master problem and to propose simpler solution methods.

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References

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

    MATH  Google Scholar 

  2. Giannessi, F., Maugeri, A. (eds.): Variational Inequalities and Network Equilibrium Problems. Plenum Press, New York (1995)

    MATH  Google Scholar 

  3. Hadjisavvas, N., Komlósi, S., Schaible, S. (eds.): Handbook of Generalized Convexity and Generalized Monotonicity. Nonconvex Optimization and Applications, vol. 76. Springer, New York (2005)

    MATH  Google Scholar 

  4. Konnov, I.V.: Equilibrium Models and Variational Inequalities. Elsevier, Amsterdam (2007)

    MATH  Google Scholar 

  5. Lasdon, L.S.: Optimization Theory for Large Systems. Macmillan, New York (1970)

    MATH  Google Scholar 

  6. Minoux, M.: Programmation Mathématique, Théorie et Algorithmes. Bordas, Paris (1989)

    Google Scholar 

  7. Hager, W.W., Hearn, D.W., Pardalos, P.M. (eds.): Large Scale Optimization: State of the Art. Kluwer, Dordrecht (1994)

    MATH  Google Scholar 

  8. Lawphongpanich, S., Hearn, D.W.: Benders decomposition for variational inequalities. Math. Program. 48, 231–247 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  9. Fuller, J., Chung, W.: Dantzig–Wolfe decomposition of variational inequalities. Comput. Econ. 25, 303–326 (2005)

    Article  MATH  Google Scholar 

  10. Fuller, J., Chung, W.: Benders decomposition for a class of variational inequalities. Eur. J. Oper. Res. 185, 76–91 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Kornai, J., Liptak, T.: Two-level planning. Econometrica 33, 141–169 (1965)

    Article  MATH  Google Scholar 

  12. Belen’kii, V.Z., Volkonskii, V.A. (eds.): Iterative Methods in Game Theory and Programming. Nauka, Moscow (1974). In Russian

    Google Scholar 

  13. Konnov, I.V.: Dual approach to one class of mixed variational inequalities. Comput. Math. Math. Phys. 42, 1276–1288 (2002)

    MathSciNet  Google Scholar 

  14. Konnov, I.V.: Application of the proximal point method to a system of extended primal-dual equilibrium problems. In: Seeger, A. (ed.) Recent Advances in Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 563, pp. 87–102. Springer, New York (2006)

    Chapter  Google Scholar 

  15. Kyparisis, J.: Sensitivity analysis for nonlinear programs and variational inequalities with nonunique multipliers. Math. Oper. Res. 15, 286–298 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kyparisis, J.: Parametric variational inequalities with multivalued solution sets. Math. Oper. Res. 17, 341–364 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  17. Yen, N.D.: Parametric optimization problems and parametric variational inequalities. Vietnam J. Math. 37, 191–223 (2009)

    MATH  MathSciNet  Google Scholar 

  18. Antipin, A.S., Vasil’ev, F.P.: A stabilization method for equilibrium programming problems with an approximately given set. Comput. Math. Math. Phys. 39, 1707–1714 (1999)

    MATH  MathSciNet  Google Scholar 

  19. Gol’shtein, E.G., Tret’yakov, N.V.: Modified Lagrange Functions. Nauka, Moscow (1989). Engl. transl. Wiley, New York (1996)

    Google Scholar 

  20. Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2001)

    Book  MATH  Google Scholar 

  21. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  22. Konnov, I.V.: On combined relaxation method’s convergence rates. Russ. Math. (Izv. VUZ) 37(12), 89–92 (1993)

    MATH  MathSciNet  Google Scholar 

  23. Konnov, I.V.: Combined relaxation methods for finding equilibrium points and solving related problems. Russ. Math. (Izv. VUZ) 37(2), 44–51 (1993)

    MATH  MathSciNet  Google Scholar 

  24. Konnov, I.V.: Combined relaxation methods for generalized monotone variational inequalities. In: Konnov, I.V., Luc, D.T., Rubinov, A.M. (eds.) Generalized Convexity and Related Topics. Lecture Notes in Economics and Mathematical Systems, vol. 583, pp. 3–31. Springer, Berlin (2007)

    Chapter  Google Scholar 

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Acknowledgements

This work was supported by grant No. 259488 from Academy of Finland and by the RFBR grant, project No. 13-01-00029a.

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  1. Department of System Analysis and Information Technologies, Kazan Federal University, ul. Kremlevskaya, 18, Kazan, 420008, Russia

    I. V. Konnov

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  1. I. V. Konnov
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Correspondence to I. V. Konnov.

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Konnov, I.V. Right-Hand Side Decomposition for Variational Inequalities. J Optim Theory Appl 160, 221–238 (2014). https://doi.org/10.1007/s10957-013-0370-0

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  • DOI: https://doi.org/10.1007/s10957-013-0370-0

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