Nonlinear Schwarz preconditioning for nonlinear optimization problems with bound constraints

TL;DR

This paper introduces a nonlinear additive Schwarz preconditioner for bound-constrained nonlinear optimization, improving scalability and performance of Newton-based methods compared to traditional active-set approaches.

Contribution

It develops a novel nonlinear Schwarz preconditioner with a solution-dependent coarse space for bound-constrained problems, enhancing algorithmic scalability and efficiency.

Findings

Preconditioned Newton methods outperform standard active-set methods.

The proposed method demonstrates improved scalability in numerical experiments.

Numerical results validate the effectiveness of the nonlinear Schwarz preconditioner.

Abstract

We propose a nonlinear additive Schwarz method for solving nonlinear optimization problems with bound constraints. Our method is used as a "right-preconditioner" for solving the first-order optimality system arising within the sequential quadratic programming (SQP) framework using Newton's method. The algorithmic scalability of this preconditioner is enhanced by incorporating a solution-dependent coarse space, which takes into account the restricted constraints from the fine level. By means of numerical examples, we demonstrate that the proposed preconditioned Newton methods outperform standard active-set methods considered in the literature.