ODYN: An All-Shifted Non-Interior-Point Method for Quadratic Programming in Robotics and AI

2026-02-17

AI Summary

This work proposes a novel non-interior-point solver for ill-posed, degenerate, and dense/sparse quadratic programming (QP) problems commonly encountered in robotics and AI, without relying on the linear independence constraint qualification. The method uniquely integrates a fully shifted nonlinear complementarity function with the proximal augmented Lagrangian approach within a sequential quadratic programming (SQP) framework, offering exceptional warm-start capabilities suited for real-time sequential optimization. The resulting implicitly differentiable QP layer, termed ODYNLayer, enables end-to-end learning and demonstrates state-of-the-art convergence performance on the Maros–Mészáros benchmark suite. The solver has been successfully deployed in model predictive control (OdynSQP), deep learning, and contact dynamics simulation (ODYNSim), confirming its versatility and computational efficiency.

Technology Category

• Intelligent Robots: Learning & Optimization for ROB
• Search and Optimization: Non-convex Optimization
• Planning, Routing, and Scheduling: Learning for Planning and Scheduling

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We introduce ODYN, a novel all-shifted primal-dual non-interior-point quadratic programming (QP) solver designed to efficiently handle challenging dense and sparse QPs. ODYN combines all-shifted nonlinear complementarity problem (NCP) functions with proximal method of multipliers to robustly address ill-conditioned and degenerate problems, without requiring linear independence of the constraints. It exhibits strong warm-start performance and is well suited to both general-purpose optimization, and robotics and AI applications, including model-based control, estimation, and kernel-based learning methods. We provide an open-source implementation and benchmark ODYN on the Maros-Mészáros test set, demonstrating state-of-the-art convergence performance in small-to-high-scale problems. The results highlight ODYN's superior warm-starting capabilities, which are critical in sequential and real-time settings common in robotics and AI. These advantages are further demonstrated by deploying ODYN as the backend of an SQP-based predictive control framework (OdynSQP), as the implicitly differentiable optimization layer for deep learning (ODYNLayer), and the optimizer of a contact-dynamics simulation (ODYNSim).

Problem

• Quadratic Programming
• Non-Interior-Point Method
• Ill-conditioned Problems
• Degenerate Problems
• Warm-start

Innovation

• all-shifted NCP
• non-interior-point method
• quadratic programming
• warm-start capability
• proximal method of multipliers