{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/on-the-differentiability-of-the-primal-dual","title":"On the Differentiability of the Primal-Dual Interior-Point Method","arxiv_id":"2406.11749","date":"2024-06-17","proceeding":null,"authors":["Kevin Tracy","Zachary Manchester"],"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.","url_abs":"https://arxiv.org/abs/2406.11749v2","url_pdf":"https://arxiv.org/pdf/2406.11749v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"on-the-differentiability-of-the-primal-dual","repo_url":"https://github.com/kevin-tracy/qpax","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"jax","reach":{"status":"ok","spdx":"NOASSERTION"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}