# Learned Preconditioning for a Primal-Dual Interior-Point Method

arXiv: https://arxiv.org/abs/2609.35665 · HTML: https://mlsanity.com/paper/2609.35665/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 21 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Builds on (most cited first, 16 of 16)

- [Learning to learn by gradient descent by gradient descent](https://mlsanity.com/paper/1606.04474/) (2016)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Exact Combinatorial Optimization with Graph Convolutional Neural Networks](https://mlsanity.com/paper/1906.01629/) (2019)
- [Learning to Optimize: A Primer and A Benchmark](https://mlsanity.com/paper/2103.12828/) (2021)
- [DC3: A learning method for optimization with hard constraints](https://mlsanity.com/paper/2104.12225/) (2021)
- [Reinforcement Learning for Integer Programming: Learning to Cut](https://mlsanity.com/paper/1906.04859/) (2019)
- [Learned Optimizers that Scale and Generalize](https://mlsanity.com/paper/1703.04813/) (2017)
- [Self-Supervised Primal-Dual Learning for Constrained Optimization](https://mlsanity.com/paper/2208.09046/) (2022)
- [Expressive Power of Graph Neural Networks for (Mixed-Integer) Quadratic Programs](https://mlsanity.com/paper/2406.05938/) (2024)
- [Towards Constituting Mathematical Structures for Learning to Optimize](https://mlsanity.com/paper/2305.18577/) (2023)
- [Learning to Pivot as a Smart Expert](https://mlsanity.com/paper/2308.08171/) (2023)
- [Self-Supervised Learning of Iterative Solvers for Constrained Optimization](https://mlsanity.com/paper/2409.08066/) (2024)
- [IPM-LSTM: A Learning-Based Interior Point Method for Solving Nonlinear Programs](https://mlsanity.com/paper/2410.15731/) (2024)
- [Projected-Search Methods for Bound-Constrained Optimization](https://mlsanity.com/paper/2110.08359/) (2021)
- [Scaling Nonlinear Optimization: Many Problems One GPU](https://mlsanity.com/paper/2606.26341/) (2026)

## Abstract (arXiv)

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.