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EXTRAGRADIENT METHODS FOR QUASI-EQUILIBRIUM PROBLEMS IN BANACH SPACES

Published online by Cambridge University Press:  01 October 2020

BEHZAD DJAFARI ROUHANI Open the ORCID record for BEHZAD DJAFARI ROUHANI [Opens in a new window]  and
VAHID MOHEBBI Open the ORCID record for VAHID MOHEBBI [Opens in a new window]
BEHZAD DJAFARI ROUHANI
Affiliation:
Department of Mathematical Sciences, University of Texas at El Paso, 500 W. University Avenue, El Paso, Texas TX 79968, USA e-mail: behzad@utep.edu
VAHID MOHEBBI
Affiliation:
Department of Mathematical Sciences, University of Texas at El Paso, 500 W. University Avenue, El Paso, Texas TX 79968, USA e-mail: vmohebbi@utep.edu
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Abstract

We study the extragradient method for solving quasi-equilibrium problems in Banach spaces, which generalizes the extragradient method for equilibrium problems and quasi-variational inequalities. We propose a regularization procedure which ensures strong convergence of the generated sequence to a solution of the quasi-equilibrium problem, under standard assumptions on the problem assuming neither any monotonicity assumption on the bifunction nor any weak continuity assumption of f in its arguments that in the many well-known methods have been used. Also, we give a necessary and sufficient condition for the solution set of the quasi-equilibrium problem to be nonempty and we show that, in this case, this iterative sequence converges strongly to a solution of the quasi-equilibrium problem. In other words, we prove strong convergence of the generated sequence to a solution of the quasi-equilibrium problem without assuming existence of a solution of the problem. Finally, we give an application of our main result to a generalized Nash equilibrium problem.

Keywords

demiclosedextragradient methodlinesearchquasi-equilibrium problemquasi ϕ-nonexpansive

MSC classification

Primary: 90C25: Convex programming
Secondary: 90C30: Nonlinear programming
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 112 , Issue 1 , February 2022 , pp. 90 - 114
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Vaithilingam Jeyakumar

References

Alber, Y. I., ‘Metric and generalized projection operators in Banach spaces: properties and applications’, in: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Applied Mathematics, 178 (Marcel Dekker, New York, 1996), 15–50.Google Scholar
Armijo, L., ‘Minimization of functions having continuous partial derivatives’, Pacific J. Math. 16 (1966), 1–3.CrossRefGoogle Scholar
Aussel, D., Cotrina, J. and Iuse, A. N., ‘An existence result for quasi-equilibrium problems’, J. Convex Anal. 24 (2017), 55–66.Google Scholar
Bianchi, M. and Schaible, S., ‘Generalized monotone bifunctions and equilibrium problems’, J. Optim. Theory Appl. 90 (1996), 31–43.CrossRefGoogle Scholar
Blum, E. and Oettli, W., ‘From optimization and variational inequalities to equilibrium problems’, Math. Student 63 (1994), 123–145.Google Scholar
Brézis, H., Nirenberg, L. and Stampacchia, S., ‘A remark on Ky Fan minimax principle’, Boll. Unione Mat. Ital. 6 (1972), 293–300.Google Scholar
Chadli, O., Chbani, Z. and Riahi, H.. ‘Equilibrium problems with generalized monotone bifunctions and applications to variational inequalities’, J. Optim. Theory Appl. 105 (2000), 299–323.Google Scholar
Combettes, P. L. and Hirstoaga, S. A., ‘Equilibrium programming in Hilbert spaces’, J. Nonlinear Convex Anal. 6 (2005), 117–136.Google Scholar
Ding, X. P., ‘Quasi-equilibrium problems with applications to infinite optimization and constrained games in general topological spaces’, Appl. Math. Lett. 13 (2000), 21–26.CrossRefGoogle Scholar
Djafari-Rouhani, B., Kazmi, K. R. and Rizvi, S. H., ‘A hybrid-extragradient-convex approximation method for a system of unrelated mixed equilibrium problems’, Trans. Math. Program. Appl. 1 (2013), 82–95.Google Scholar
Fan, K., ‘A minimax inequality and applications’, in: Inequality III (ed. Shisha, O.) (Academic Press, New York, 1972), 103–113.Google Scholar
Golshtein, E. G. and Tretyakov, N. V., Modified Lagrangians and Monotone Maps in Optimization (John Wiley, New York, 1996).Google Scholar
Iusem, A. N., Kassay, G. and Sosa, W., ‘On certain conditions for the existence of solutions of equilibrium problems’, Math. Program. 116 (2009), 259–273.CrossRefGoogle Scholar
Iusem, A. N. and Lucambio-Pérez, L. R., ‘An extragradient-type algorithm for non-smooth variational inequalities’, Optimization 48 (2000), 309–332.CrossRefGoogle Scholar
Iusem, A. N. and Mohebbi, V., ‘Extragradient methods for nonsmooth equilibrium problems in Banach spaces’, Optimization (to appear). Published online (2 May 2018).Google Scholar
Iusem, A. N. and Mohebbi, V., ‘Extragradient methods for vector equilibrium problems in Banach spaces’, Numer. Funct. Anal. Optim. 40 (2019), 993–1022.CrossRefGoogle Scholar
Iusem, A. N. and Nasri, M.. ‘Korpolevich’s method for variational inequality problems in Banach spaces’, J. Global Optim. 50 (2011), 59–76.CrossRefGoogle Scholar
Iusem, A. N. and Sosa, W., ‘Iterative algorithms for equilibrium problems’, Optimization 52 (2003), 301–316.CrossRefGoogle Scholar
Iusem, A. N. and Sosa, W., ‘On the proximal point method for equilibrium problems in Hilbert spaces’, Optimization 59 (2010), 1259–1274.CrossRefGoogle Scholar
Iusem, A. N. and Svaiter, B. F., ‘A variant of Korpelevich’s method for variational inequalities with a new search strategy’, Optimization 42 (1997), 309–321.CrossRefGoogle Scholar
Kamimura, S. and Takahashi, W., ‘Strong convergence of a proximal-type algorithm in a Banach space’, SIAM J. Optim. 13 (2002), 938–945.CrossRefGoogle Scholar
Khobotov, E. N., ‘Modifications of the extragradient method for solving variational inequalities and certain optimization problems’, USSR Comput. Math. Math. Phys. 27 (1987), 120–127.CrossRefGoogle Scholar
Konnov, I. V., ‘Combined relaxation methods for finding equilibrium points and solving related problems’, Russian Math. 37 (1993), 34–51.Google Scholar
Korpelevich, G. M., ‘The extragradient method for finding saddle points and other problems’, Ekonom. Mat. Metody 12 (1976), 747–756 (in Russian).Google Scholar
Marcotte, P., ‘Application of Khobotov’s algorithm to variational inequalities and network equilibrium problems’, INFOR Inf. Syst. Oper. Res. 29 (1991), 258–270.Google Scholar
Moudafi, A., ‘Proximal point algorithm extended to equilibrium problems’, J. Nat. Geom. 15 (1999), 91–100.Google Scholar
Muu, L. D., Nguyen, V. H. and Quy, N. V., ‘On Nash–Cournot oligopolistic market equilibrium models with concave cost functions’, J. Global Optim. 41 (2008), 351–364.CrossRefGoogle Scholar
Pang, J. S. and Fukushima, M., ‘Quasi-variational inequalities, generalized Nash equilibria and multi-leader–follower games’, Comput. Manag. Sci. 2 (2005), 21–56.CrossRefGoogle Scholar
Phelps, R. R., ‘Lectures on maximal monotone operators’, Extracta Math. 12 (1997), 193–230.Google Scholar
Quoc, T. D., Anh, P. N. and Muu, L. D., ‘Dual extragradient algorithms extended to equilibrium problems’, J. Global Optim. 52 (2012), 139–159.CrossRefGoogle Scholar
Quoc, T. D., Muu, L. D. and Nguyen, V. H., ‘Extragradient algorithm extended to equilibrium problems’, Optimization 57 (2008), 749–776.CrossRefGoogle Scholar
Reich, S., ‘A weak convergence theorem for the alternating method with Bregman distances’, in: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type , Lecture Notes in Pure and Applied Mathematics, 178 (Marcel Dekker, New York, NY, 1996), 313–318.Google Scholar
Van, N. T. T., Strodiot, J. J., Nguyen, V. H. and Vuong, P. T., ‘An extragradient-type method for solving nonmonotone quasi-equilibrium problems’, Optimization 67 (2018), 651–664.CrossRefGoogle Scholar