Preconditioned Latent Variable Proximal Point Method for Optimal Control Problems with Control Constraints
Abstract
The latent variable proximal point (LVPP) method solves optimization problems in function spaces with pointwise constraints by a sequence of smooth subproblems whose iterates are strictly feasible. We apply it to linear-quadratic elliptic optimal control problems. Control and observation are distributed but may be restricted to subdomains. We prove that the LVPP subproblems are well posed in appropriate function spaces, that the iteration converges for every sequence of proximity parameters with divergent sum, and that it delivers the Lagrange multiplier of the constraint as a by-product. We interpret the Newton step for the LVPP subproblem as a linear-quadratic optimal control problem with a spatially varying control cost, and we show that Newton's method converges locally quadratically. The Newton systems are solved by the minimal residual method with a block-diagonal preconditioner. We prove that the spectrum of the Newton operator with respect to the inner product defined by the preconditioner is bounded independently of the proximity parameter, the iterate, the mesh and the observation and control domains. The results are combined into a complete algorithm, and tested in numerical experiments.
Related
- A Preconditioned Semismooth Newton Method for Box-Constrained Elliptic Optimal Control Problems with Random Coefficients
- Anchored Extra-Proximal Methods: Optimal Higher-Order Methods for Monotone Inclusion Problems
- An Inexact Halpern-accelerated Preconditioned Generalized Proximal Point Algorithm for the Maximal Monotone Inclusion Problem
- Resilient Decentralized Preconditioned Stochastic Proximal Optimization with Gradient Sketching over Time-Varying Networks
- Preconditioned Three-Term Conjugate Gradient Framework for Nonconvex Finite Minimax Problems
BibTeX
@misc{haaf2026preconditioned,
title = {{Preconditioned Latent Variable Proximal Point Method for Optimal Control Problems with Control Constraints}},
author = {Nico Haaf and Roland Herzog},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16678},
eprint = {2610.16678},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16678},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Preconditioned Latent Variable Proximal Point Method for Optimal Control Problems with Control Constraints
AU - Nico Haaf
AU - Roland Herzog
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16678
UR - https://arxiv.org/abs/2610.16678
AB - The latent variable proximal point (LVPP) method solves optimization problems in function spaces with pointwise constraints by a sequence of smooth subproblems whose iterates are strictly feasible. We apply it to linear-quadratic elliptic optimal control problems. Control and observation are distributed but may be restricted to subdomains. We prove that the LVPP subproblems are well posed in appropriate function spaces, that the iteration converges for every sequence of proximity parameters with divergent sum, and that it delivers the Lagrange multiplier of the constraint as a by-product. We interpret the Newton step for the LVPP subproblem as a linear-quadratic optimal control problem with a spatially varying control cost, and we show that Newton's method converges locally quadratically. The Newton systems are solved by the minimal residual method with a block-diagonal preconditioner. We prove that the spectrum of the Newton operator with respect to the inner product defined by the preconditioner is bounded independently of the proximity parameter, the iterate, the mesh and the observation and control domains. The results are combined into a complete algorithm, and tested in numerical experiments.
ER -