An overlapping waveform-relaxation preconditioner for economic optimal control problems with state constraints

TL;DR

This paper introduces an overlapping waveform-relaxation preconditioner for solving parabolic economic optimal control problems with state constraints, improving the convergence robustness of Newton methods as the regularization parameter decreases.

Contribution

A novel nonlinear preconditioner based on an overlapping optimized waveform-relaxation method with Robin conditions is proposed for better convergence in constrained optimal control problems.

Findings

Preconditioned Newton method shows robust convergence with respect to the regularization parameter.

Proper overlap and Robin parameter choices enhance solver performance.

Numerical experiments validate the effectiveness of the proposed preconditioner.

Abstract

In this work, a class of parabolic economic optimal control problems is considered. These problems are characterized by pointwise state constraints regularized by a parameter, which transforms the pure state constraints in mixed control-state ones. However, the convergence of classical (semismooth) Newton methods deteriorates for decreasing values of the regularization parameter. To tackle this problem, a nonlinear preconditioner is introduced. This is based on an overlapping optimized waveform-relaxation method characterized by Robin transmission conditions. Numerical experiments show that appropriate choices of the overlap and of the Robin parameter lead to a preconditioned Newton method with a robust convergence against the state constraints regularization parameter.