A new iterative method for generalized equilibrium and constrained convex minimization problems

M. Yazdi


The gradient-projection algorithm (GPA) plays an important role in solving constrained convex minimization problems. In this paper, we combine the GPA and averaged mapping approach to propose an explicit composite iterative scheme for finding a common solution of a generalized equilibrium problem and a constrained convex minimization problem. Then, we prove a strong convergence theorem which improves and extends some recent results.


Generalized equilibrium problem; constrained convex minimization; averaged mapping; iterative method; variational inequality

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Aoyama, K., Kimura, Y., Takahashi, W., Toyoda, M., Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2007), 2350–2360.

Blum, E., Oettli, W., From optimization and variatinal inequalities to equilibrium problems, Math. Student. 63 (1994), 123–145.

Bertsekas, D. P., Gafin, E. M., Projection methods for variational inequalities with applications to the traffic assignment problem, Math. Program. Stud. 17 (1982), 139–159.

Byrne, C., A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Probl. 20 (2004), 103–120.

Combettes, P. L., Hirstoaga, S. A., Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005), 117–136.

Goebel, K., Kirk, W. A., Topics in Metric Fixed Point Theory, Cambridge Stud. Adv. Math. 28, Cambridge Univ. Press, Cambridge, 1990.

Han, D., Lo, H. K., Solving non-additive traffic assignment problems: A descent method for co-coercive variational inequalities, Eur. J. Oper. Res. 159 (2004), 529–544.

Jung, J. S., A general composite iterative method for equilibrium problems and fixed point problems, J. Comput. Anal. Appl. 12 (1-A) (2010), 124–140.

Moudafi, A., Thera, M., Proximal and dynamical approaches to equilibrium problems, in: Ill-posed Variational Problems and Regularization Techniques, Lecture Notes in Economics and Mathematical Systems, 477, Springer-Verlag, Berlin, 1999, 187–201.

Peng, J.-W., Yao, J.-Ch., A viscosity approximation scheme for system of equilibrium problems, nonexpansive mappings and monotone mappings, Nonlinear Anal. 71 (12) (2009), 6001–6010.

Plubtieg, S., Punpaeng, R., A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 336 (2007), 455–469.

Razani, A., Yazdi, M., Viscosity approximation method for equilibrium and fixed point problems, Fixed Point Theory. 14 (2) (2013), 455–472.

Tian, M., Liu, L., Iterative algorithms based on the viscosity approximation method for equilibrium and constrained convex minimization problem, Fixed Point Theory Appl. 2012:201 (2012), 1–17.

Wang, S., Hu, Ch., Chai, G., Strong convergence of a new composite iterative method for equilibrium problems and fixed point problems, Appl. Math. Comput. 215 (2010), 3891–3898.

Xu, H.-K., Iterative algorithms for nonlinear operators, J. London Math. Soc. (2) 66 (2002), 240–256.

Xu, H.-K., An iterative approach to quadratic optimization, J. Optim. Theory Appl.116 (2003), 659–678.

Xu, H.-K., Averaged mappings and the gradient-projection algorithm, J. Optim. Theory Appl. 150 (2011), 360–378.

Yazdi, M., New iterative methods for equilibrium and constrained convex minimization problems, Asian-Eur. J. Math. 12 (3) (2019), 12 pp.

DOI: http://dx.doi.org/10.17951/a.2020.74.2.81-99
Data publikacji: 2020-12-28 17:42:03
Data złożenia artykułu: 2020-12-27 17:39:57


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