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

Jose Rojas; Aristotelis Papatheodorou; Sergi Martinez; Andrea Patrizi; Ioannis Havoutis; Carlos Mastalli

cs.ROFeb 17, 2026

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

Authors: Jose Rojas, Aristotelis Papatheodorou, Sergi Martinez, Andrea Patrizi, Ioannis Havoutis, Carlos Mastalli

Organizations: Robot Motor Intelligence (RoMI) Lab, Heriot-Watt University, U.K. · Oxford Robotics Institute (ORI), University of Oxford, U.K. · Humanoids and Human Centered Mechatronics, Italian Institute of Technology, Italy.

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 & tables

Fig. 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.

Method	Solver	Algorithm	Backend	Warm-start	Degeneracy	Accuracy	Differentiable
Non-interior Point	Odyn	All-shifted NIPM	Dense / Sparse	✓	✓	High	✓
	Fbstab [ 32 ]	Semismooth NIPM	Sparse	✓	✓	High	✗
Interior-Point	PiQP [ 9 ]	Proximal IPM	Sparse	✗	✓	High	✗
	Mosek [ 8 ]	Homogeneous self-dual IPM	Dense / Sparse	✗	✓	High	✗
	Gurobi [ 7 ]	Primal–dual IPM	Dense / Sparse	✗	✓	High	✗
	qpSWIFT [ 22 ]	Primal–dual IPM	Sparse	✗	✓	High	✗

TABLE I : Comparison of representative QP solvers grouped by algorithmic method.

Fig. 2 : Log-barrier function as

\mu

approaches

0

. The approximation becomes closer to the indicator function.

Parameter	Value	Description
$\beta$ , $\theta$	$0.85,0.95$	neighborhood hyper-parameters
$\sigma_{\text{min}}$ , $\sigma_{\text{max}}$	$0.1,0.9$	minimum and maximum centering
$\mu_{\text{min}}$	$10^{-16}$	minimum barrier value
$\theta_{\text{min}}$	$0.1$	minimum interpolation value
$\theta_{l}$ , $\theta_{u}$	$0.3,0.85$	min. and max. improvement thresholds
$\theta^{-}$ , $\theta^{+}$	$0.4,0.3$	decrease and increase constants

TABLE II : Odyn ’s hyper-parameters for double precision.

Accuracy	Odyn	ProxQP	PiQP	OsQP	Mosek	Gurobi
Medium	9.7%	33.0%	10.7%	44.7%	14.6%	9.7%
High	15.5%	35.9%	10.7%	49.5%	14.6%	12.6%

TABLE III : Failure rate (%) of different QP solvers on a subset of the Maros–Mészáros test set at medium and high accuracy.

Fig. 3 : (top) Iteration based performance profiles at high accuracy. (bottom) Solve time performance profiles at high accuracy.

Fig. 4 : Warm-to-cold ratio performance profiles under high-accuracy settings, evaluated with perturbed QP data. The top and bottom plots correspond to perturbation levels

\delta=10^{-3}

and

10^{-1}

, respectively.

Fig. 5 : Performance of the dense backend of Odyn , ProxQP , and PiQP . We generate random QP 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.

Solver	Redundant Ineq.	LICQ Failure	Multi. Solutions
Odyn	5	10	6
ProxQP	12	30	10
PiQP	5	8	6
OsQP	75	$\dagger$	200
Mosek	4	14	5
Gurobi	11	8	9

TABLE IV : Number of iterations required by each solver on the three degenerate QP problems. We highlight the three best solvers.

Fig. 6 : Snapshots of optimal trajectories computed with OdynSQP . (Top) G1 humanoid performing a backflip under joint position, velocity, and torque limits, together with contact wrench-cone constraints. (Bottom-left) Borinot aerial robot reaching a target end-effector position (green sphere) while satisfying actuator torque limits. (Bottom-right) B1–Z1 executing a chicken-head stabilization task during lateral locomotion while enforcing end-effector and friction-cone constraints; the robot’s hand is constrained to remain within the green bounding box.

Fig. 7 : Feasibility evolution on challenging constrained optimal control problems computed by OdynSQP .

Fig. 8 : Snapshots of MPC trajectories generated by OdynSQP . (Top) B1 quadruped executing a quasi-static walking-in-place motion under contact and friction-cone constraints, while enforcing state and input constraints. (Bottom) Z1 manipulator tracking a three-dimensional ellipsoidal end-effector trajectory (blue dashed curve) under state and input constraints. Despite changing active sets and high-frequency updates, OdynSQP achieved accurate tracking and stable behavior across successive MPC iterations.To watch the video, click the picture or see https://www.youtube.com/watch?v=fJqjgdT2De0 .

Fig. 9 : Model predictive control tracking performance on the Unitree B1 quadruped using OdynSQP . The plots show the tracking of the center-of-mass position (bottom) and angular momentum (top) along the

y

-axis during a quasi-static walking-in-place motion.

Fig. 10 : Model predictive control tracking performance on the Unitree Z1 manipulator using OdynSQP . The plots show the tracking of the end-effector position along selected axes while following a three-dimensional ellipsoidal trajectory.

Panel of Figure 15 — Fig. 11 : Pushed cube contact simulation resolved via QP -based contact dynamics. (Top) illustrate the cube’s interaction with the plane under unilateral frictional contact constraints at different time instants. (Bottom) compares the number of solver iterations required per time step for Odyn , ProxQP , and OsQP .

Panel of Figure 16 — Fig. 12 : Effect of ODYNLayer precision on convergence and invalid-Sudoku rate. (a) With double precision ( float64 ), it exhibited stable convergence with a rapid decline in invalid Sudoku solutions. (b) With single precision ( float32 ), it showed slightly slower convergence in terms of invalid Sudoku count, a degradation that is attributed to Odyn ’s single-precision convergence tolerance and randomized-parameter initialization. (c) Example reconstruction comparing ground truth with the model prediction; digits shown in blue denote the fixed givens supplied to ODYNLayer . These results highlight how numerical precision impacts both convergence speed and true Sudoku constraint satisfaction (invalid Sudoku count). True Sudoku constraint satisfaction implies that the learned linear constraint basis accurately captures the underlying Sudoku structure.

		Double ( float64 )		Float ( float32 )
		#errors	iters ( $\pm\sigma$ )	#errors	iters ( $\pm\sigma$ )
Full	Dense	0	$11.38\;(\pm 0.31)$	0	$8.91\;(\pm 0.30)$
Full	Sparse	0	$11.16\;(\pm 0.29)$	0	$8.98\;(\pm 0.30)$
Cond.	Dense	0	$11.36\;(\pm 0.28)$	- (max iters)	- (max iters)
Cond.	Sparse	0	$11.14\;(\pm 0.28)$	- (max iters)	- (max iters)

TABLE V : Sudoku ablation study with ODYNLayer : We indicate the average solver iterations per epoch (iters) during training and the number of invalid Sudoku predictions (#errors) in the out-of-distribution test-set (

1000

Sudoku instances). The full- KKT factorization setup is indicated by “Full” and the condensed by “Cond.”.

Quadratic Programming

Quadratic programming optimizes quadratic objectives under constraints, supporting applications from robotics and control to finance and machine learning.

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

Aristotelis Papatheodorou, Jose Rojas, Ioannis Havoutis, Carlos Mastalli

Department of Engineering Science, University of Oxford, Oxford, UK · Robot Motor Intelligence (RoMI) Lab, Heriot-Watt University, Edinburgh, UK

Robotic Control Primal-Dual Optimization Trajectory Optimization Constrained Optimization Quadratic Programming Differentiable Optimization

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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-

\ell_2

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.

Abstract

Quadratic Programming

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