Content area
Full Text
http://crossmark.crossref.org/dialog/?doi=10.1007/s10957-015-0833-6&domain=pdf
Web End = http://crossmark.crossref.org/dialog/?doi=10.1007/s10957-015-0833-6&domain=pdf
Web End = http://crossmark.crossref.org/dialog/?doi=10.1007/s10957-015-0833-6&domain=pdf
Web End = http://crossmark.crossref.org/dialog/?doi=10.1007/s10957-015-0833-6&domain=pdf
Web End = http://crossmark.crossref.org/dialog/?doi=10.1007/s10957-015-0833-6&domain=pdf
Web End = J Optim Theory Appl (2016) 168:756768 DOI 10.1007/s10957-015-0833-6
http://crossmark.crossref.org/dialog/?doi=10.1007/s10957-015-0833-6&domain=pdf
Web End = http://crossmark.crossref.org/dialog/?doi=10.1007/s10957-015-0833-6&domain=pdf
Web End = http://crossmark.crossref.org/dialog/?doi=10.1007/s10957-015-0833-6&domain=pdf
Web End = http://crossmark.crossref.org/dialog/?doi=10.1007/s10957-015-0833-6&domain=pdf
Web End = Extended Lorentz Cones and Variational Inequalities on Cylinders
Sndor Zoltn Nmeth1 Guohan Zhang1
Received: 26 May 2015 / Accepted: 22 October 2015 / Published online: 6 November 2015 The Author(s) 2015. This article is published with open access at Springerlink.com
Abstract Solutions of a variational inequality problem dened by a closed and convex set and a mapping are found by imposing conditions for the monotone convergence with respect to a cone of the Picard iteration corresponding to the composition of the projection onto the dening closed and convex set and the difference in the identity mapping and the dening mapping. One of these conditions is the isotonicity of the projection onto the dening closed and convex set. If the closed and convex set is a cylinder and the cone is an extented Lorentz cone, then this condition can be dropped because it is automatically satised. In this case, a large class of afne mappings and cylinders which satisfy the conditions of monotone convergence above is presented. The obtained results are further specialized for unbounded box-constrained variational inequalities. In a particular case of a cylinder with a base being a cone, the variational inequality is reduced to a generalized mixed complementarity problem which has been already considered in Nmeth and Zhang (J Global Optim 62(3):443457, 2015).
Keywords Isotone projections Cones Variational inequalities Picard iteration
Fixed point
Mathematics Subject Classication 90C33 47H07 47H99 47H09
B Sndor Zoltn Nmeth
s.nemeth@bham.ac.uk
Guohan Zhang gxz245@bham.ac.uk
1 School of Mathematics, University of Birmingham, Watson Building,
Edgbaston, Birmingham B15 2TT, UK
123
J Optim Theory Appl (2016) 168:756768 757
1 Introduction
In this paper, we will study the solvability of variational inequalities on closed and convex sets by using the isotonicity of the metric projection mapping onto these sets with respect to the partial order dened by a cone. Apparently, this approach has not been considered before.
Variational inequalities are models of various important problems in physics, engineering, economics, and other sciences. The classical Nash equilibrium concept can...