Extended Lorentz cones and variational inequalities on cylinders

Solutions of a variational inequality problem defined 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 defining closed and convex set and the difference in the identity mapping and the defining mapping. One of these conditions is the isotonicity of the projection onto the defining 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 satisfied. In this case, a large class of affine 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 Németh and Zhang (J Global Optim 62(3):443–457, 2015).