Learned Preconditioning for a Primal-Dual Interior-Point Method
Abstract
Interior-point methods (IPMs) are among the most widely used algorithms for constrained optimization, yet their Newton-based search directions require costly second-order information and large linear-system solves. Learning to optimize offers cheaper updates learned from data, but the singular behavior of logarithmic barriers near constraint boundaries makes IPMs highly sensitive to perturbations, complicating both warm starting and learning reliable updates. We introduce pdLIP, an IPM for smooth nonlinear programs that integrates learned preconditioning with pdProj, an all-shifted primal-dual projected-search IPM. A shared coordinate-wise recurrent network predicts a positive diagonal preconditioner that scales the right-hand side of the reduced Newton system for the primal step, and the remaining slack and multiplier directions are recovered analytically. The learned iterations avoid Hessian evaluations and Newton-system solves, using only first-order and coordinate-wise operations amenable to GPU parallelization. Training is self-supervised, with a loss based on a penalty-barrier merit function and the residual of perturbed optimality conditions, requiring neither target directions nor precomputed solutions. Primal and dual shifts mitigate the barrier's sensitivity to perturbations near constraint boundaries, enabling effective warm starting. Across four classes of 200-dimensional convex and nonconvex constrained problems, pdLIP warm starts reduce pdProj refinement iterations by 63-67% compared with cold starts at the same KKT residual tolerance of 10−8, with negligible warm-start generation cost relative to the subsequent pdProj solve. Improvements persist on box-constrained QPs with 1000 variables and extend to applications including portfolio optimization, support vector machines, and a nonlinear control example.
Related
- A Retrospective-Approximation Interior-Point Method
- A Practical Combinatorial Interior Point Method for Minimum-Cost Flow
- A parameterized block LU preconditioning for double saddle point problems
- A unified variational framework for augmented Lagrangian preconditioning of saddle point systems: applications to mixed-dimensional and fictitious domain methods
- Accelerated primal-dual dynamics and algorithms for strongly convex-concave bilinear saddle point problems
BibTeX
@misc{madabhushi2026learned,
title = {{Learned Preconditioning for a Primal-Dual Interior-Point Method}},
author = {Abhinav Madabhushi and Jialin Liu and Minxin Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35665},
eprint = {2609.35665},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35665},
}
RIS
TY - UNPB
TI - Learned Preconditioning for a Primal-Dual Interior-Point Method
AU - Abhinav Madabhushi
AU - Jialin Liu
AU - Minxin Zhang
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35665
UR - https://arxiv.org/abs/2609.35665
AB - Interior-point methods (IPMs) are among the most widely used algorithms for constrained optimization, yet their Newton-based search directions require costly second-order information and large linear-system solves. Learning to optimize offers cheaper updates learned from data, but the singular behavior of logarithmic barriers near constraint boundaries makes IPMs highly sensitive to perturbations, complicating both warm starting and learning reliable updates. We introduce pdLIP, an IPM for smooth nonlinear programs that integrates learned preconditioning with pdProj, an all-shifted primal-dual projected-search IPM. A shared coordinate-wise recurrent network predicts a positive diagonal preconditioner that scales the right-hand side of the reduced Newton system for the primal step, and the remaining slack and multiplier directions are recovered analytically. The learned iterations avoid Hessian evaluations and Newton-system solves, using only first-order and coordinate-wise operations amenable to GPU parallelization. Training is self-supervised, with a loss based on a penalty-barrier merit function and the residual of perturbed optimality conditions, requiring neither target directions nor precomputed solutions. Primal and dual shifts mitigate the barrier's sensitivity to perturbations near constraint boundaries, enabling effective warm starting. Across four classes of 200-dimensional convex and nonconvex constrained problems, pdLIP warm starts reduce pdProj refinement iterations by 63-67% compared with cold starts at the same KKT residual tolerance of 10^−8, with negligible warm-start generation cost relative to the subsequent pdProj solve. Improvements persist on box-constrained QPs with 1000 variables and extend to applications including portfolio optimization, support vector machines, and a nonlinear control example.
ER -