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Dual extragradient algorithms extended to equilibrium problems

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Abstract

In this paper we propose two iterative schemes for solving equilibrium problems which are called dual extragradient algorithms. In contrast with the primal extragradient methods in Quoc et al. (Optimization 57(6):749–776, 2008) which require to solve two general strongly convex programs at each iteration, the dual extragradient algorithms proposed in this paper only need to solve, at each iteration, one general strongly convex program, one projection problem and one subgradient calculation. Moreover, we provide the worst case complexity bounds of these algorithms, which have not been done in the primal extragradient methods yet. An application to Nash-Cournot equilibrium models of electricity markets is presented and implemented to examine the performance of the proposed algorithms.

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References

  1. Blum E., Oettli W.: From optimization and variational inequality to equilibrium problems. Math. Student 63, 127–149 (1994)

    Google Scholar 

  2. Bruck R.E.: On the weak convergence of an Ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space. J. Math. Anal. Appl. 61, 159–164 (1977)

    Article  Google Scholar 

  3. Chinchuluun, A., Pardalos, P.M., Migdalas, A., Pitsoulis, L. (eds): Pareto Optimality, Game Theory and Equilibria. Springer, Berlin (2008)

    Google Scholar 

  4. Cohen G.: Auxiliary problem principle and decomposition of optimization problems. J. Optim. Theory Appl. 32, 277–305 (1980)

    Article  Google Scholar 

  5. Cohen G.: Auxiliary principle extended to variational inequalities. J. Optim. Theory Appl. 59, 325–333 (1988)

    Article  Google Scholar 

  6. Contreras J., Klusch M., Krawczyk J.B.: Numerical solutions to Nash-Cournot equilibria in coupled constraint electricity markets. IEEE Trans. Power Syst. 19(1), 195–206 (2004)

    Article  Google Scholar 

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

    Google Scholar 

  8. Flam S.D., Antipin A.S.: Equilibrium programming using proximal-like algorithms. Math. Program. 78, 29–41 (1997)

    Article  Google Scholar 

  9. Giannessi, F., Maugeri, A., Pardalos, P.M. (eds): Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models. Kluwer, Dordrecht (2004)

    Google Scholar 

  10. Konnov I.V.: Combined relaxation methods for variational inequalities. Springer-Verlag, Berlin (2001)

    Book  Google Scholar 

  11. Konnov I.V., Kum S.: Descent methods for mixed variational inequalities in Hilbert spaces. Nonlinear Anal. 47, 561–572 (2001)

    Article  Google Scholar 

  12. Konnov I.V.: Generalized convexity and related topics. In: Konnov, I.V., Luc, D.T., Rubinov, (eds.) A.M. (eds) Combined Relaxation Methods for Generalized Monotone Variational Inequalities, pp. 3–331. Springer, Berlin (2007)

    Google Scholar 

  13. Korpelevich G.M.: Extragradient method for finding saddle points and other problems. Matecon 12, 747–756 (1976)

    Google Scholar 

  14. Lalitha C.S.: A note on duality of generalized equilibrium problem. Optim. Lett. 4(1), 57–66 (2010)

    Article  Google Scholar 

  15. Li S.J., Zhao P.: A method of duality for mixed vector equilibrium problem. Optim. Lett. 4(1), 85–96 (2010)

    Article  Google Scholar 

  16. Maiorano, A., Song, Y.H., Trovato, M.: Dynamics of noncollusive oligopolistic electricity markets. In: Proceedings IEEE Power Engineering Society Winter Meeting, pp. 838–844, Singapore Jan (2000)

  17. Martinet B.: Régularisation d’inéquations variationelles par approximations successives. Revue Française d’Automatique et d’Informatique Recherche Opérationnelle 4, 154–159 (1970)

    Google Scholar 

  18. Mastroeni G.: On auxiliary principle for equilibrium problems. Publicatione del Dipartimento di Mathematica dellUniversita di Pisa 3, 1244–1258 (2000)

    Google Scholar 

  19. Mastroeni G.: Gap function for equilibrium problems. J. Global Optim. 27(4), 411–426 (2003)

    Article  Google Scholar 

  20. Moudafi A.: Proximal point algorithm extended to equilibrium problem. J. Nat. Geom. 15, 91–100 (1999)

    Google Scholar 

  21. Muu L.D., Oettli W.: Convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlinear Anal. 18(12), 1159–1166 (1992)

    Article  Google Scholar 

  22. Muu L.D., Quoc T.D.: Regularization algorithms for solving monotone Ky Fan inequalities with application to a Nash-Cournot equilibrium model. J. Optim. Theory Appl. 142(1), 185–204 (2009)

    Article  Google Scholar 

  23. Nemirovskii A.S.: Effective iterative methods for solving equations with monotone operators. Ekon. Matem. Met. (Matecon) 17, 344–359 (1981)

    Google Scholar 

  24. Nesterov Y.: Dual extrapolation and its applications to solving variational inequalities and related problems. Math. Program. Ser. B 109(2–3), 319–344 (2007)

    Article  Google Scholar 

  25. Nguyen V.H.: Lecture Notes on Equilibrium Problems. CIUF-CUD Summer School on Optimization and Applied Mathematics. Nha Trang, Vietnam (2002)

    Google Scholar 

  26. Panicucci B., Pappalardo M., Passacantando M.: On solving generalized Nash equilibrium problems via optimization. Optim. Lett. 3(3), 419–435 (2009)

    Article  Google Scholar 

  27. Pardalos, P.M, Rassias, T.M., Khan, A.A. (eds): Nonlinear Analysis and Variational Problems. Springer, Berlin (2010)

    Google Scholar 

  28. Quoc T.D., Muu L.D.: Implementable quadratic regularization methods for solving pseudomonotone equilibrium problems. East West J. Math. 6(2), 101–123 (2004)

    Google Scholar 

  29. Quoc T.D., Muu L.D., Nguyen V.H.: Extragradient algorithms extended to equilibrium problems. Optimization 57(6), 749–776 (2008)

    Article  Google Scholar 

  30. Rockafellar R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Google Scholar 

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

    Article  Google Scholar 

  32. Taskar B., Lacoste-Julien S., Jordan M.I.: Structured prediction, dual extragradient and Bregman projections. J. Mach. Learn. Res. 7, 1627–1653 (2006)

    Google Scholar 

  33. Van N.T.T., Strodiot J.J., Nguyen V.H.: The interior proximal extragradient method for solving equilibrium problems. J. Global Optim. 44(2), 175–192 (2009)

    Article  Google Scholar 

  34. Van N.T.T., Strodiot J.J., Nguyen V.H.: A bundle method for solving equilibrium problems. Math. Program. 116, 529–552 (2009)

    Article  Google Scholar 

  35. Wachter A., Biegler L.T.: On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006)

    Article  Google Scholar 

  36. Zhu D.L., Marcotte P.: An extended descent framework for variational inequalities. J. Optim. Theory Appl. 80, 349–366 (1994)

    Article  Google Scholar 

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Author notes

  1. Tran D. Quoc

    Present address: Department of Electrical Engineering (ESAT/SCD) and OPTEC, K.U. Leuven, Leuven, Belgium

Authors and Affiliations

  1. Hanoi University of Science, Hanoi, Vietnam

    Tran D. Quoc

  2. Posts and Telecommunications Institute of Technology, Hanoi, Vietnam

    Pham N. Anh

  3. Institute of Mathematics, Hanoi, Vietnam

    Le D. Muu

Authors
  1. Tran D. Quoc
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  2. Pham N. Anh
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  3. Le D. Muu
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Correspondence to Tran D. Quoc.

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Quoc, T.D., Anh, P.N. & Muu, L.D. Dual extragradient algorithms extended to equilibrium problems. J Glob Optim 52, 139–159 (2012). https://doi.org/10.1007/s10898-011-9693-2

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  • DOI: https://doi.org/10.1007/s10898-011-9693-2

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