On Non-Reducible Multi-Player Control Problems and their Numerical Computation

TL;DR

This paper studies a class of multi-player Nash equilibrium control problems that are non-reducible, proposing semi-smooth Newton and active-set methods with superlinear convergence, supported by finite element discretizations and numerical validation.

Contribution

It introduces a novel class of non-reducible multi-player control problems and develops superlinear convergent semi-smooth Newton and active-set algorithms for their solution.

Findings

Superlinear convergence of the proposed Newton and active-set methods.

Finite element discretizations effectively implement the algorithms.

Numerical examples confirm theoretical convergence results.

Abstract

In this article we consider a special class of Nash equilibrium problems that cannot be reduced to a single player control problem. Problems of this type can be solved by a semi-smooth Newton method. Applying results from the established convergence analysis we derive superlinear convergence for the associated Newton method and the equivalent active-set method. We also provide detailed finite element discretizations for both methods. Several numerical examples are presented to support the theoretical findings.