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Nonlinear Schwarz preconditioning for nonlinear optimization problems with bound constraints

Hardik Kothari, Alena Kopaničáková, Rolf Krause

TL;DR

This paper addresses bound-constrained nonlinear optimization problems arising from finite element discretizations by introducing nonlinear additive Schwarz preconditioners as right preconditioners for Newton-SQP. It presents two variants, NRAS-B and TL-NRAS-B, which solve constrained subproblems on overlapping subdomains and incorporate a coarse-space correction with a first-order consistent augmented coarse objective $\hat{f}_0$ to ensure global information transfer. The two-level method employs an inverted V-cycle to combine coarse corrections with subdomain updates, preserving feasibility throughout. Numerical experiments on ignition and minimal-surface problems show that the preconditioned Newton-SQP methods outperform standard active-set approaches, with the two-level TL-NRAS-B method offering superior scalability when the coarse space adequately captures fine-level constraints.

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.

Nonlinear Schwarz preconditioning for nonlinear optimization problems with bound constraints

TL;DR

This paper addresses bound-constrained nonlinear optimization problems arising from finite element discretizations by introducing nonlinear additive Schwarz preconditioners as right preconditioners for Newton-SQP. It presents two variants, NRAS-B and TL-NRAS-B, which solve constrained subproblems on overlapping subdomains and incorporate a coarse-space correction with a first-order consistent augmented coarse objective to ensure global information transfer. The two-level method employs an inverted V-cycle to combine coarse corrections with subdomain updates, preserving feasibility throughout. Numerical experiments on ignition and minimal-surface problems show that the preconditioned Newton-SQP methods outperform standard active-set approaches, with the two-level TL-NRAS-B method offering superior scalability when the coarse space adequately captures fine-level constraints.

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.
Paper Structure (4 sections, 12 equations, 2 figures, 1 algorithm)

This paper contains 4 sections, 12 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Nonlinear Preconditioning
  3. Numerical Experiments
  4. Conclusion

Figures (2)

  • Figure 1: Convergence history of NRAS-B (Left) and TL-NRAS-B (Right) methods for the ignition problem (Top) and the minimal surface (Bottom) problem. The experiments are performed with an increasing number of subdomains (sbd).
  • Figure 2: Convergence history of semismooth Newton (SS-Newton), Newton-SQP, RASPN-B, and TL-RASPN-B methods for the ignition problem (Left) and minimal surface problem (Right).