On the application of the SCD semismooth* Newton method to variational inequalities of the second kind

TL;DR

This paper develops and applies the SCD semismooth* Newton method to variational inequalities of the second kind, resulting in an efficient, locally superlinear convergent algorithm and globally convergent hybrid methods demonstrated on large-scale Cournot-Nash equilibrium problems.

Contribution

The paper introduces an implementable SCD semismooth* Newton method for variational inequalities of the second kind, with globally convergent hybrid algorithms and practical efficiency demonstrated on large-scale equilibrium models.

Findings

The method achieves locally superlinear convergence.

Hybrid algorithms improve global convergence.

Efficiently solves large-scale Cournot-Nash equilibrium problems.

Abstract

The paper starts with a description of the SCD (subspace containing derivative) mappings and the SCD semismooth* Newton method for the solution of general inclusions. This method is then applied to a class of variational inequalities of the second kind. As a result, one obtains an implementable algorithm exhibiting a locally superlinear convergence. Thereafter we suggest several globally convergent hybrid algorithms in which one combines the SCD semismooth* Newton method with selected splitting algorithms for the solution of monotone variational inequalities. Finally we demonstrate the efficiency of one of these methods via a Cournot-Nash equilibrium, modeled as a variational inequalities of the second kind, where one admits really large numbers of players (firms) and produced commodities.