# Elastic ODYN: Differentiable Optimization for Infeasible Control and Learning in Robotics

arXiv: https://arxiv.org/abs/2606.16564 · HTML: https://mlsanity.com/paper/2606.16564/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 39 references that are in the map.
- First posted on arXiv 15 Jun 2026.

## Builds on (most cited first, 17 of 17)

- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [OSQP: An Operator Splitting Solver for Quadratic Programs](https://mlsanity.com/paper/1711.08013/) (2017)
- [End-to-End Constrained Optimization Learning: A Survey](https://mlsanity.com/paper/2103.16378/) (2021)
- [Conservative set valued fields, automatic differentiation, stochastic gradient method and deep learning](https://mlsanity.com/paper/1909.10300/) (2019)
- [Alternating Differentiation for Optimization Layers](https://mlsanity.com/paper/2210.01802/) (2022)
- [Flexible Differentiable Optimization via Model Transformations](https://mlsanity.com/paper/2206.06135/) (2022)
- [BPQP: A Differentiable Convex Optimization Framework for Efficient End-to-End Learning](https://mlsanity.com/paper/2411.19285/) (2024)
- [PIQP: A Proximal Interior-Point Quadratic Programming Solver](https://mlsanity.com/paper/2304.00290/) (2023)
- [Agile Maneuvers in Legged Robots: a Predictive Control Approach](https://mlsanity.com/paper/2203.07554/) (2022)
- [Contact-Implicit Model Predictive Control: Controlling Diverse Quadruped Motions Without Pre-Planned Contact Modes or Trajectories](https://mlsanity.com/paper/2312.08961/) (2023)
- [Constrained Differential Dynamic Programming: A primal-dual augmented Lagrangian approach](https://mlsanity.com/paper/2210.15409/) (2022)
- [On the Differentiability of the Primal-Dual Interior-Point Method](https://mlsanity.com/paper/2406.11749/) (2024)
- [FBstab: A Stabilized Semismooth Quadratic Programming Algorithm with Applications in Model Predictive Control](https://mlsanity.com/paper/1901.04046/) (2019)
- [A Regularized and Smoothed Fischer-Burmeister Method for Quadratic Programming with Applications to Model Predictive Control](https://mlsanity.com/paper/1807.03214/) (2018)
- [Momentum-Aware Trajectory Optimisation using Full-Centroidal Dynamics and Implicit Inverse Kinematics](https://mlsanity.com/paper/2310.06074/) (2023)
- [ODYN: An All-Shifted Non-Interior-Point Method for Quadratic Programming in Robotics and AI](https://mlsanity.com/paper/2602.16005/) (2026)

## Abstract (arXiv)

Robotic systems routinely encounter conflicting objectives, modeling errors, and degenerate contact conditions that render quadratic programs (QPs) infeasible. Yet most optimization solvers and differentiable QP layers assume feasibility, leading to numerical failures, unstable gradients, or solver breakdown when constraints cannot be simultaneously satisfied. We present Elastic ODYN, a primal-dual non-interior-point QP solver that handles infeasibility through smooth squared-ℓ2 elastic relaxations. The formulation remains well posed under ill-conditioning and degeneracy, supports warm starting, and converges to closest-to-feasible solutions, with lightweight refinement recovering physically meaningful dual variables. Building on this framework, we develop Elastic ODYNLayer, a differentiable QP layer with stable gradients under infeasibility, and Elastic OdynSQP, an SQP method that resolves inconsistent subproblems and intrinsically infeasible optimal control tasks through selective constraint elasticity. Across benchmark QPs, singular contact mechanics, differentiable parameter identification, and quadrupedal and humanoid trajectory optimization, Elastic ODYN outperforms state-of-the-art elastic QP solvers in robustness, warm-start performance, and convergence reliability, enabling optimization, simulation, control, and learning beyond standard feasibility assumptions.