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The Proximal Point Method for Nonmonotone Variational Inequalities

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Abstract

We consider an application of the proximal point method to variational inequality problems subject to box constraints, whose cost mappings possess order monotonicity properties instead of the usual monotonicity ones. Usually, convergence results of such methods require the additional boundedness assumption of the solutions set. We suggest another approach to obtaining convergence results for proximal point methods which is based on the assumption that the dual variational inequality is solvable. Then the solutions set may be unbounded. We present classes of economic equilibrium problems which satisfy such assumptions.

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References

  • Bakushinsky AB, Goncharsky AV (1994) Ill-posed problems: theory and applications. Kluwer, Dordrecht

    Google Scholar 

  • Billups SC, Ferris MC, (1997) QPCOMP: a quadratic programming based solver for mixed complementarity problems. Math Progr 76:533–562

    MathSciNet  Google Scholar 

  • Crouzeix J-P (1997) Pseudomonotone variational inequality problems: existence of solutions. Math Progr 78:305–314

    MathSciNet  Google Scholar 

  • Daniilidis A, Hadjisavvas N (1999) Coercivity conditions and variational inequalities. Math Progr 86:433–438

    Article  MathSciNet  MATH  Google Scholar 

  • Daniilidis A, Hadjisavvas N (1999) Characterization of nonsmooth semistrictly quasiconvex and strictly quasiconvex functions. J Optim Theory Appl 102:525–536

    Article  MathSciNet  MATH  Google Scholar 

  • Eckstein J, Bertsekas DP (1992) On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math Progr 55:293–318

    Article  MathSciNet  MATH  Google Scholar 

  • El Farouq N (2001) Pseudomonotone variational inequalities: convergence of proximal methods. J Optim Theory Appl 109:311–326

    Article  MathSciNet  MATH  Google Scholar 

  • Facchinei F, Kanzow C (1999) Beyond monotonicity in regularization methods for linear complementarity problems. SIAM J Contr Optim 37:1150–1161

    Article  MathSciNet  MATH  Google Scholar 

  • Facchinei F, Pang J-S (2003) Finite-dimensional variational inequalities and complementarity problems. Springer, Berlin Heidelberg, New York

    Google Scholar 

  • Fang SC, Petersen EL (1982) Generalized variational inequalities. J Optim Theory Appl 38: 363–383

    Article  MathSciNet  MATH  Google Scholar 

  • Flam SD, Antipin AS (1997) Equilibrium programming using proximal algorithms. Math Progr 78:29–41

    Article  MathSciNet  Google Scholar 

  • Gol’shtein EG, Tret’yakov NV (1996) Modified lagrangians and monotone maps in optimization. J Wiley, New York

    Google Scholar 

  • Konnov IV (1996) A general approach to finding stationary points and the solution of related problems. Comp Math Math Phys 36:585–593

    MathSciNet  MATH  Google Scholar 

  • Konnov IV (1998). On quasimonotone variational inequalities. J Optim Theory Appl 99:165–181

    Article  MathSciNet  MATH  Google Scholar 

  • Konnov IV (2000) Properties of gap functions for mixed variational inequalities. Siberian J Numer Math 3:259–270

    MATH  Google Scholar 

  • Konnov IV (2001) Combined relaxation methods for variational inequalities. Springer Berlin heidelberg, New York

    MATH  Google Scholar 

  • Konnov (2003) Application of the proximal point method to non monotone equilibrium problems. J Optim Theory Appl 119:317–333

    Article  MathSciNet  MATH  Google Scholar 

  • Konnov IV, Volotskaya EO (2002) Mixed variational inequalities and ecomonic equilibrium problems. J Appl Math 2:289–314

    Article  MathSciNet  MATH  Google Scholar 

  • Martinet B (1970) Regularization d’inéquations variationnelles par approximations successives. Rev Fr d’Inform Rech Opér 4:154–159

    MathSciNet  Google Scholar 

  • Moré J, Rheinboldt W (1973) On P- and S-functions and related classes of n-dimensional nonlinear mappings. Linear Algebra Appl 6:45–68

    Article  MATH  Google Scholar 

  • Nikaido H (1968) Convex structures and economic theory. Academic, New York

    MATH  Google Scholar 

  • Patriksson M (1999) Nonlinear programming and variational inequality problems: a unified approach. Kluwer Dordrecht

  • Polterovich VM, Spivak VA (1982) Mappings with gross substitutability in the theory of economic equilibrium. In: Gamkrelidze RV (ed) Contemporary mathematics problems, 19 (in Russian). VINITI, Moscow, pp 111–154

  • Polterovich VM, Spivak VA (1983) Gross substitutability of point-to-set correspondences. J Math Econ 11:117–140

    Article  MathSciNet  MATH  Google Scholar 

  • Qi HD (1999) Tikhonov regularization methods for variational inequality problems. J Optim Theory Appl 102:193–201

    Article  MathSciNet  MATH  Google Scholar 

  • Rockafellar RT (1976) Monotone operators and the proximal point algorithm. SIAM J Contrib Optim 14:877–898

    Article  MathSciNet  MATH  Google Scholar 

  • Saigal R (1976) Extensions of the generalized complementarity problem. Math Oper Res 1:260–266

    Article  MathSciNet  MATH  Google Scholar 

  • Shih MH, Tan KK (1988) Browder–Hartmann–Stampacchia variational inequalities for multivalued monotone operators. J Math Anal Appl 134:431–440

    Article  MathSciNet  MATH  Google Scholar 

  • Yamashita N, Imai J, Fukushima M (2001) The proximal point algorithm for the P 0 complementarity problem. In: Ferris MC, Mangasarian OL, Pang J-S (eds) Complementarity: applications, algorithms, and extensions. Kluwer, Dordrecht, pp 361–379

    Google Scholar 

  • Yao JC (1994) Multi-valued variational inequalities with K-pseudomonotone operators. J Optim Theory Appl 83:391–403

    Article  MathSciNet  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Quantitative Methods, Brescia University, Contrada Santa Chiara, 50, Brescia, 25122, Italy

    E. Allevi

  2. Department of Mathematics, Statistics, Informatics and Applications, Bergamo University, Via dei Caniana, 2, Bergamo, 24127, Italy

    A. Gnudi

  3. Department of Applied Mathematics, Kazan University, ul. Kremlevskaya,18, Kazan, 420008, Russia

    I. V. Konnov

Authors
  1. E. Allevi
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  2. A. Gnudi
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  3. I. V. Konnov
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Correspondence to I. V. Konnov.

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Allevi, E., Gnudi, A. & Konnov, I.V. The Proximal Point Method for Nonmonotone Variational Inequalities. Math Meth Oper Res 63, 553–565 (2006). https://doi.org/10.1007/s00186-005-0052-2

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  • DOI: https://doi.org/10.1007/s00186-005-0052-2

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