# On the Differentiability of the Primal-Dual Interior-Point Method

arXiv: https://arxiv.org/abs/2406.11749 · HTML: https://mlsanity.com/paper/2406.11749/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Jun 2024.

## Built on by (most cited first, 6 of 6)

- [MPAX: Mathematical Programming in JAX](https://mlsanity.com/paper/2412.09734/) (2024)
- [Learning responsibility allocations for multi-agent interactions: A differentiable optimization approach with control barrier functions](https://mlsanity.com/paper/2410.07409/) (2024)
- [Learning Probabilistic Responsibility Allocations for Multi-Agent Interactions](https://mlsanity.com/paper/2604.13128/) (2026)
- [Provably Safe, Yet Scalable Reinforcement Learning](https://mlsanity.com/paper/2606.14536/) (2026)
- [Constrained Whole-Body Tracking for Humanoid Robots](https://mlsanity.com/paper/2606.00374/) (2026)
- [Elastic ODYN: Differentiable Optimization for Infeasible Control and Learning in Robotics](https://mlsanity.com/paper/2606.16564/) (2026)

## Abstract (arXiv)

Primal-Dual Interior-Point methods are capable of solving constrained convex optimization problems to tight tolerances in a fast and robust manner. The derivatives of the primal-dual solution with respect to the problem matrices can be computed using the implicit function theorem, enabling efficient differentiation of these optimizers for a fraction of the cost of the total solution time. In the presence of active inequality constraints, this technique is only capable of providing discontinuous subgradients that present a challenge to algorithms that rely on the smoothness of these derivatives. This paper presents a technique for relaxing primal-dual solutions with a logarithmic barrier to provide smooth derivatives near active inequality constraints, with the ability to specify a uniform and consistent amount of smoothing. We pair this with an efficient primal-dual interior-point algorithm for solving an always-feasible ℓ1-penalized variant of a convex quadratic program, eliminating the issues surrounding learning potentially infeasible problems. This parallelizable and smoothly differentiable solver is demonstrated on a range of robotics tasks where smoothing is important. An open source implementation in JAX is available at github.com/kevin-tracy/qpax.