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Ceng, Lu-Chuan; Liao, Cheng-Wen; Pang, Chin-Tzong; Wen, Ching-Feng

Hybrid iterative scheme for triple hierarchical variational inequalities with mixed equilibrium, variational inclusion, and minimization constraints. (English) Zbl 1473.47030

Abstr. Appl. Anal. 2014, Article ID 767109, 22 p. (2014).
Summary: We introduce and analyze a hybrid iterative algorithm by combining Korpelevich’s extragradient method, the hybrid steepest-descent method, and the averaged mapping approach to the gradient-projection algorithm. It is proven that, under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of finitely many nonexpansive mappings, the solution set of a generalized mixed equilibrium problem (GMEP), the solution set of finitely many variational inclusions, and the solution set of a convex minimization problem (CMP), which is also a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm to solving a hierarchical variational inequality problem with constraints of the GMEP, the CMP, and finitely many variational inclusions.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
49J40 Variational inequalities
47J22 Variational and other types of inclusions

Keywords:

strong convergence; hybrid iterative algorithm; hybrid steepest-descent method; gradient-projection algorithm; real Hilbert space
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\textit{L.-C. Ceng} et al., Abstr. Appl. Anal. 2014, Article ID 767109, 22 p. (2014; Zbl 1473.47030)
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