In this paper we propose an algorithm for solving the split feasibility problem \(x\in C, Ax\in Q\) with C being the solution set of an equilibrium problem and A can be nonlinear. The proposed algorithm is a combination between the projection method for the equilibrium problem and the gradient method for the inclusion \(Ax\in Q\). The convergence of the algorithm is investigated. A numerical example for a jointly constrained Nash equilibrium model in electricity production market is provided to demonstrate the behavior of the algorithm.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Alber, Y.I., Iusem, A.N., Solodov, M.V.: On the projected subgradient method for nonsmooth convex optimization in a Hilbert space. Math. Program. 81, 23–35 (1998)
Anh, T.V., Muu, L.D.: A projection-fixed point method for a class of bilevel variational inequalities with split fixed point constraints. Optimization 65, 1129–1143 (2016)
Aussel, D., Bendottib, P., Pitek, M.: Nash equilibriumin a pay-as-bid electricity market Part 2—best response of a producer. Optimization 66, 1027–1053 (2017)
Avriel, M.: Nonlinear Programming: Analysis and Methods. Prentice-Hall, Englewood Cliffs (1976)
Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, Berlin (2011)
Bigi, G., Castellani, M., Pappalardo, M., Panssacantando, M.: Existence and solution methods for equilibria. Eur. Oper. Res. 227, 1–11 (2013)
Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 127–149 (1994)
Bnouhachem, A., Noor, M.A., Khalfaoui, M., Zhaohan, S.: On descent-projection method for solving the split feasibility problems. J. Glob. Optim. 54, 627–639 (2012)
Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problems. Inverse Prob. 18, 441–453 (2002)
Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Prob. 20, 103–120 (2004)
Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006)
Censor, Y., Elfving, T.: A multiprojections algorithm using Bregman projections in a product spaces. Numer. Algorithms 8, 221–239 (1994)
Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Prob. 21, 2071–2084 (2005)
Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms 59, 301–323 (2012)
Censor, Y., Segal, A.: The split common fixed point problems for directed operators. J. Convex Anal. 16, 587–600 (2009)
Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM, Philadelphia (1983)
Contreras, J., Klusch, M., Krawczyk, J.B.: Numerical solution to Nash–Cournot equilibria in coupled constraint electricity markets. EEE Trans. Power Syst. 19, 195–206 (2004)
Deepho, J., Kumam, W., Kumam, P.: A new hybrid projection algorithm for solving the split generalized equilibrium problems and the system of variational inequality problems. J. Math. Model. Algorithms 13, 405423 (2014)
Dragomir, S.S., Pearce, C.E.M.: Jensen’s inequality for quasiconvex functions. Numer. Algebra Control Optim. 2(2), 279–291 (2012)
Finetti, D., Sulle, B.: Stratificazioni conversse. Ann. Mat. Pura Appl. 30, 173–183 (1949)
Giorgi, G.: A simple way to prove the characterization of differentiable quasiconvex functions. Appl. Math. 5, 1226–1228 (2014)
Gorbachuk, V.M.: Cournot–Nash and Bertrand–Nash equilibria for heterogeneous duopoly of differentiated products. Cyber. Syst. Anal. 46, 25–26 (2010)
Hieu, D.V.: Two hybrid algorithms for solving split equilibrium problems. Int. J. Comput. Math. 95(3), 561–583 (2018)
Iusem, A.N.: On some properties of paramonotone operator. Convex Anal. 5, 269–278 (1998)
Jing-Yuan, W., Smeers, Y.: Spatial oligopolistic electricity models with Cournot generators and regulated transmission prices. Oper. Res. 47, 102112 (1999)
Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2000)
Kiwiel, K.C.: Convergence and efficiency of subgradient methods for quasiconvex minimization. Math. Program. Ser. A 90, 1–25 (2001)
Kraikaew, R., Saejung, S.: On split common fixed point problems. J. Math. Anal. Appl. 415, 513–524 (2014)
Krawczyk, J.B., Uryasev, S.: Relaxation algorithms to find Nash equilibria with economic applications. Environ. Model. Assess. 5, 63–73 (2000)
Li, Z., Hana, D., Zhang, W.: Self-adaptive projection-type method for nonlinear multiple-sets split feasibility problem. Inverse Probl. Sci. Eng. 21, 155–170 (2013)
Lopez, G., Martin-Maquez, V., Wang, F., Xu, H.K.: Solving the split feasibility problem without prior knowledge of matrix norms. Inverse Prob. 28, 085–094 (2012)
Luc, D.L.: Characterisations of quasiconvex functions. Bull. Aust. Math. Soc. 48, 393–406 (1993)
Moudafi, A.: Split monotone variational inclusions. J. Optim. Theory Appl. 150, 275–283 (2011)
Moudafi, A., Thakur, B.S.: Solving proximal split feasibility problems without prior knowledge of operator norms. Optim. Lett. 8, 2099–2110 (2014)
Muu, L.D., Oettli, W.: Convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlinear Anal. 18, 1159–1166 (1992)
Santos, P., Scheimberg, S.: An inexact subgradient algorithm for equilibrium problems. Comput. Appl. Math. 30, 91–107 (2011)
Santos, P., Scheimberg, S.: A modified projection algorithm for constrained equilibrium problems. Optimization 66(12), 2051–2062 (2017)
Shehu, Y., Ogbuisi, F.U.: Convergence analysis for proximal split feasibility problems and fixed point problems. J. Appl. Math. Comput. 48, 221–239 (2015)
Vuong, P.T., Strodiot, J.J., Nguyen, V.H.: A gradient projection method for solving split equality and split feasibility problems in Hilbert spaces. Optimization 64, 2321–2341 (2015)
Yen, L.H., Muu, L.D., Huyen, N.T.T.: An algorithm for a class of split feasibility problems: application to a model in electricity production. Math. Methods Oper. Res. 84(3), 549–565 (2016)
We would like to thank the editor and the referees very much for their useful comments, remarks and suggestions that helped us very much to improve quality of the paper. The first author would also like to thank the Vietnam Institute for Advanced Study in Mathematics (VIASM) for the support during her visit.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This paper is supported by the NAFOSTED, Grant 101.01-2017.315.
About this article
Cite this article
Yen, L.H., Huyen, N.T.T. & Muu, L.D. A subgradient algorithm for a class of nonlinear split feasibility problems: application to jointly constrained Nash equilibrium models. J Glob Optim 73, 849–868 (2019). https://doi.org/10.1007/s10898-018-00735-0
- Nonlinear split feasibility
- Subgradient method
- Nash model