Hybrid implicit steepest-descent methods for triple hierarchical variational inequalities with hierarchical variational inequality constraints

Volume 10, Issue 7, pp 3963--3987 http://dx.doi.org/10.22436/jnsa.010.07.47

Publication Date: July 27, 2017 Submission Date: June 09, 2017

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Authors

Lu-Chuan Ceng - Department of Mathematics, Shanghai Normal University, and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China. Yeong-Cheng Liou - Department of Healthcare Administration and Medical Informatics, and Research Center of Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung 807, Taiwan. - Department of Medical Research, Kaohsiung Medical University Hospital, Kaohsiung 807, Taiwan. Ching-Feng Wen - Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 80708, Taiwan. Ching-Hua Lo - Department of Management, Yango University, Fujian 350015, China.

Abstract

In this paper, we introduce and analyze a hybrid implicit steepest-descent algorithm for solving the triple hierarchical variational inequality problem with the hierarchical variational inequality constraint for finitely many nonexpansive mappings in a real Hilbert space. The proposed algorithm is based on Korpelevich’s extragradient method, hybrid steepest-descent method, Mann’s implicit iteration method, and Halpern’s iteration method. Under mild conditions, the strong convergence of the iteration sequences generated by the algorithm is established. Our results improve and extend the corresponding results in the earlier and recent literature.

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Keywords

• Hybrid implicit steepest-descent algorithm
• triple hierarchical variational inequality
• Mann’s implicit iteration method
• nonexpansive mapping
• inverse-strong monotonicity
• global convergence.

MSC

•  49J30
•  47H09
•  47J20
•  49M05

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