Triple Hierarchical Variational Inequalities with Constraints of Mixed Equilibria, Variational Inequalities, Convex Minimization, and Hierarchical Fixed Point Problems | Zendy

Lu-Chuan Ceng | Zendy; Cheng-Wen Liao | Zendy; Chin-Tzong Pang | Zendy; ChingFeng Wen | Zendy

Open Access

Triple Hierarchical Variational Inequalities with Constraints of Mixed Equilibria, Variational Inequalities, Convex Minimization, and Hierarchical Fixed Point Problems

Author(s) -

Lu-Chuan Ceng,

Cheng-Wen Liao,

Chin-Tzong Pang,

ChingFeng Wen

Publication year - 2014

Publication title -

journal of applied mathematics

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.307

H-Index - 43

eISSN - 1687-0042

pISSN - 1110-757X

DOI - 10.1155/2014/129379

Subject(s) - variational inequality , mathematics , fixed point , hilbert space , solution set , regular polygon , projection (relational algebra) , minification , mathematical optimization , gradient descent , set (abstract data type) , convex set , convex optimization , algorithm , pure mathematics , computer science , mathematical analysis , artificial neural network , geometry , machine learning , programming language

We introduce and analyze a hybrid iterative algorithm by virtue of Korpelevich's extragradient method, viscosity approximation method, hybrid steepest-descent method, and 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 infinitely many nonexpansive mappings, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inequality problems (VIPs), the solution set of general system of variational inequalities (GSVI), and the set of minimizers of convex minimization problem (CMP), which is just 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 solve a hierarchical fixed point problem with constraints of finitely many GMEPs, finitely many VIPs, GSVI, and CMP. The results obtained in this paper improve and extend the corresponding results announced by many others

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