Extended Lorentz cones and variational inequalities on cylinders

TL;DR

This paper investigates variational inequalities on cylindrical sets using extended Lorentz cones, showing that certain monotonicity conditions can be automatically satisfied, with applications to box-constrained problems and a numerical example.

Contribution

It introduces conditions under which the monotone convergence of Picard iterations is guaranteed for variational inequalities on cylinders with extended Lorentz cones, simplifying existing assumptions.

Findings

Monotone convergence conditions are automatically satisfied for cylinders with extended Lorentz cones.

The approach applies to unbounded box constrained variational inequalities.

A numerical example demonstrates the practical application of the theoretical results.

Abstract

Solutions of a variational inequality are found by giving conditions for the monotone convergence with respect to a cone of the Picard iteration corresponding to its natural map. One of these conditions is the isotonicity of the projection onto the closed convex set in the definition of the variational inequality. If the closed convex set is a cylinder and the cone is an extented Lorentz cone, then this condition can be dropped because it is automatically satisfied. The obtained result is further particularized for unbounded box constrained variational inequalities. For this case a numerical example is presented.