### Extragradient method for convex minimization problem

Lu-Chuan Ceng, Yeong-Cheng Liou, Ching-Feng Wen
2014 Journal of Inequalities and Applications
In this paper, we introduce and analyze a multi-step hybrid extragradient algorithm by combining Korpelevich's extragradient method, the viscosity approximation method, the hybrid steepest-descent method, Mann's iteration method and the gradient-projection method (GPM) with regularization in the setting of infinite-dimensional Hilbert spaces. It is proven that, under appropriate assumptions, the proposed algorithm converges strongly to a solution of the convex minimization problem (CMP) with
more » ... straints of several problems: finitely many generalized mixed equilibrium problems (GMEPs), finitely many variational inclusions, and the fixed point problem of a strictly pseudocontractive mapping. In the meantime, we also prove the strong convergence of the proposed algorithm to the unique solution of a hierarchical variational inequality problem (over the fixed point set of a strictly pseudocontractive mapping) with constraints of finitely many GMEPs, finitely many variational inclusions and the CMP. The results presented in this paper improve and extend the corresponding results announced by many others. MSC: 49J30; 47H09; 47J20; 49M05 Keywords: hybrid extragradient approach; split feasibility problem; generalized mixed equilibrium problem; variational inclusion; strictly pseudocontractive mapping; nonexpansive mapping In terms of Huang [] (see also []), we have the following property for the resolvent operator J R,λ : H → D(R). Lemma . J R,λ is single-valued and firmly nonexpansive, i.e., Consequently, J R,λ is nonexpansive and monotone. Lemma . (see []) Let R be a maximal monotone mapping with D(R) = C. Then for any given λ > , u ∈ C is a solution of problem (.) if and only if u ∈ C satisfies u = J R,λ (u -λBu). Lemma . (see []) Let R be a maximal monotone mapping with D(R) = C and let B : C → H be a strongly monotone, continuous, and single-valued mapping. Then for each z ∈ H, the equation z ∈ (B + λR)x has a unique solution x λ for λ > . Lemma . (see []) Let R be a maximal monotone mapping with D(R) = C and B : C → H be a monotone, continuous and single-valued mapping. Then (I + λ(R + B))C = H for each λ > . In this case, R + B is maximal monotone.