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

TL;DR

This paper introduces a smoothing technique for primal-dual interior-point methods that enables differentiability near active inequality constraints, facilitating efficient and smooth optimization in robotics applications.

Contribution

It presents a novel smoothing approach for primal-dual solutions with a logarithmic barrier, improving differentiability and robustness in convex quadratic programming.

Findings

Enables smooth derivatives near active constraints

Efficient primal-dual interior-point algorithm for $\,\ell_1$-penalized QPs

Demonstrated on robotics tasks with open-source JAX implementation

Abstract

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 $\ell_1$-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.