Hybrid steepest-descent methods for systems of variational inequalities with constraints of variational inclusions and convex minimization problems

Volume 10, Issue 3, pp 874--901 http://dx.doi.org/10.22436/jnsa.010.03.03

Publication Date: March 20, 2017 Submission Date: November 28, 2016

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Authors

Zhao-Rong Kong - Economics Management Department, Shanghai University of Political Science and Law, Shanghai 201701, China. 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 for Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung 807, Taiwan. Ching-Feng Wen - Center for Fundamental Science, and Research Center for Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung, 80708, Taiwan. - Department of Medical Research, Kaohsiung Medical University Hospital, Kaohsiung 807, Taiwan.

Abstract

Two hybrid steepest-descent schemes (implicit and explicit) for finding a solution of the general system of variational inequalities (in short, GSVI) with the constraints of finitely many variational inclusions for maximal monotone and inversestrongly monotone mappings and a minimization problem for a convex and continuously Fréchet differentiable functional (in short, CMP) have been presented in a real Hilbert space. We establish the strong convergence of these two hybrid steepestdescent schemes to the same solution of the GSVI, which is also a common solution of these finitely many variational inclusions and the CMP. Our results extend, improve, complement and develop the corresponding ones given by some authors recently in this area.

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Keywords

• Hybrid steepest-descent method
• system of variational inequalities
• variational inclusion
• monotone mapping.

MSC

•  49J30
•  47H09
•  47J20
•  49M05

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