A projection-type method for variational inequalities on Hadamard manifolds and verification of solution existence

Abstract

In this paper, we extend a projection-type method for variational inequalities from Euclidean spaces to Hadamard manifolds. The proposed method has the following nice features: (i) the algorithm is well defined whether the solution set of the problem is non-empty or not, under weak assumptions; (ii) if the solution set is non-empty, then the sequence generated by the method is convergent to the solution, which is closest to the initial point; and (iii) the existence of the solutions to variational inequalities can be verified through the behaviour of the generated sequence. The results presented in this paper generalize and improve some known results given in literatures.

Keywords:

• extragradient method
• variational inequality
• Hadamard manifold
• pseudomonotone vector field

AMS Subject Classifications:

• 47J20
• 47J25
• 47H05
• 58C30

Additional information

Funding

This work is supported by the Natural Science Foundation of P.R. China [grant numbers 10962001 and 51149007], Guangxi Natural Science Foundation [grant numbers 2013GXNSFBA019015, 2011GXNSFD018006, and 2010GXNSFA013115], Scientific Research Foundation of Guangxi Universities [grant number 201102ZD014], Scientific Research Foundation of Guangxi University for Nationalities [grant number 2012QD015] and Open fund of Guangxi key laboratory of hybrid computation and IC design analysis [grant number 2013HCIC06].