A unified hybrid iterative method for hierarchical minimization problems

In this paper, we introduce and analyze a new unified hybrid iterative method to compute the approximate solution of the general optimization problem defined over the set D=Fix(T)∩Ω[GMEP(Φ,Ψ,φ)], where Fix(T)Fix(T) is the set of common fixed points of a family T={T(t):0≤t<∞}T={T(t):0≤t<∞} of nonexpansive self-mappings on a Hilbert space HH, and Ω[GMEP(Φ,Ψ,φ)] is the set of solutions of the generalized mixed equilibrium problem (in short, GMEP). Such type of minimization problem is called the hierarchical minimization problem. We establish the strong convergence of the sequences generated by the proposed algorithm. Our strong convergence theorem extends, improves and unifies the previously known results in the literature. We also give a numerical example to illustrate our algorithm and results.