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

TL;DR

ODYN is introduced, a novel all-shifted primal-dual non-interior-point quadratic programming (QP) solver designed to efficiently handle challenging dense and sparse QPs and its superior warm-starting capabilities are highlighted.

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).

Figures (9)

• Figure 1: Overview of Odyn applications in robotics and AI. Odyn serves as the computational core for constrained nonlinear trajectory optimization (OdynSQP), contact-dynamics simulation (ODYNSim), and differentiable optimization layers (ODYNLayer), providing a common optimization backbone across control, simulation, and learning.
• Figure 2: Log-barrier function as $\mu$ approaches $0$. The approximation becomes closer to the indicator function.
• Figure 3: (top) Iteration based performance profiles at high accuracy. (bottom) Solve time performance profiles at high accuracy.
• Figure 4: Warm-to-cold ratio performance profiles under high-accuracy settings, evaluated with perturbed data. The top and bottom plots correspond to perturbation levels $\delta = 10^{-3}$ and $10^{-1}$, respectively.
• Figure 5: Performance of the dense backend of Odyn, ProxQP, and PiQP. We generate random problems with dimensions ranging from $1$ to $400$ and report the average computation time per iteration over $10$ trials. As expected, all solvers exhibit cubic complexity, whereas Odyn and PiQP demonstrate more deterministic timing behaviour.