Infeasible optimization problems and the hierarchical augmented Lagrangian method in imitation learning

Ajay Sathya; Justin Carpentier; Roland Andrews

Infeasible optimization problems and the hierarchical augmented Lagrangian method in imitation learning

T0 review · 0 major / 1 minor · reviewed 2026-06-28 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A hierarchical augmented Lagrangian method drives imitation learning policies to the closest feasible constrained solution even when hard constraints are initially infeasible.

desk verdict Applies hierarchical ALM to steer IL toward closest-feasible solutions when constraints conflict, with consistent toy validation but narrow novelty beyond the application. read the letter →

arxiv 2606.00730 v1 pith:ZIGJJPCZ submitted 2026-05-30 cs.RO

Roland Andrews , Justin Carpentier , Ajay Sathya This is my paper · ORCID

classification cs.RO

keywords imitationlearningaugmentedLagrangianmethodinfeasibleconstraintsconstrainedoptimizationroboticspoliciessafetyhierarchicalformulationclosestfeasiblesolution

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Imitation learning for robotics policies often adds hard constraints to enforce safety and stability, but these constraints can prove infeasible and produce unstable or difficult training. The paper studies a remedy that applies recent theory on the augmented Lagrangian method for infeasible problems through a hierarchical formulation. This steers the learned policy to the solution of the closest-feasible constrained imitation learning problem, which retains desirable properties. The approach is tested on a toy driving task with total-acceleration and pedestrian-safety constraints, where infeasibility arises naturally yet a safe policy remains possible. A sympathetic reader cares because the method supplies a stable training pathway when perfect satisfaction of all constraints cannot be achieved.

What carries the argument

The hierarchical augmented Lagrangian method, which reformulates the constrained imitation learning problem to converge on the closest feasible solution under infeasible constraints.

What would settle it

Apply the method to an imitation learning problem whose constraints are known to be infeasible and check whether the policy converges to the one minimizing constraint violation while matching the expert demonstrations as closely as possible; failure to stabilize or to reach that closest solution would falsify the claim.

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Extended reading notes

Core claim

We show that our approach drives the learned policy toward the solution of a closest-feasible constrained IL problem with desirable properties. The method is illustrated on a toy driving example with a total-acceleration constraint and pedestrian-safety constraints, a setting in which infeasibility can naturally arise while still allowing a safe learned policy.

Load-bearing premise

Recent theoretical results on the augmented Lagrangian method in infeasible settings can be directly applied via a hierarchical formulation to produce stable training dynamics and a closest-feasible solution in imitation learning problems.

Editorial extensions

If this is right

The learned policy solves the closest-feasible constrained IL problem rather than failing outright.
Training dynamics stay stable despite the presence of infeasible constraints.
The method produces safe policies in concrete settings such as driving with acceleration and pedestrian-safety constraints.
The hierarchical formulation inherits the convergence properties established by recent theory for infeasible augmented Lagrangians.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same hierarchical treatment could be tested on other constrained learning settings outside imitation learning where infeasibility occurs.
Robotics practitioners might adopt the method to reduce manual constraint tuning when safety requirements conflict.
Comparing the closest-feasible policy against policies obtained by ad-hoc constraint softening would quantify any advantage in policy quality.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The main point is that this paper takes recent results on the augmented Lagrangian method for infeasible problems and puts them into a hierarchical formulation for imitation learning. When hard constraints on the policy cannot all be met, the method still drives the solution toward the closest feasible version rather than producing unstable training.

They lay out the issue clearly with robotics examples where safety and dynamics constraints can clash, then show the hierarchical penalty structure maps to the desired closest-feasible outcome. The toy driving case with total-acceleration and pedestrian-safety constraints illustrates the setting without obvious gaps in the logic or hidden assumptions in the parameterization.

The derivation holds up internally and the example is reproducible enough to check the claim. No free parameters or invented entities appear in the setup, and the citation pattern stays grounded in the referenced ALM theory.

The soft spot is that the core guarantee comes from prior work, so the contribution is mainly the targeted adaptation to IL rather than new theory or broad empirical tests. The validation stays at the toy level, which leaves open how well it scales or compares to simpler relaxation methods.

This is useful for people working on constrained imitation learning in robotics who hit feasibility walls. It shows honest engagement with the literature and a testable claim, so it deserves a serious referee.

Referee Report

0 major / 1 minor

Summary. The paper argues that hard constraints in imitation learning (IL) optimization problems can be infeasible, leading to unstable training dynamics. It proposes a hierarchical augmented Lagrangian method (ALM) based on recent theoretical results for infeasible settings, claiming this drives the learned policy to the solution of a closest-feasible constrained IL problem with desirable properties. The approach is illustrated on a toy driving example involving a total-acceleration constraint and pedestrian-safety constraints.

Significance. If the central claim holds, the work offers a theoretically grounded remedy for infeasibility in constrained IL, which is common in robotics applications where safety and stability constraints may conflict. It extends recent ALM theory to hierarchical formulations in policy learning, potentially enabling more stable training while preserving safety properties in the closest-feasible solution. The toy example provides initial validation in a setting where infeasibility arises naturally.

minor comments (1)

The abstract and introduction would benefit from a brief explicit statement of the hierarchical formulation (e.g., how the outer and inner ALM levels are structured) to make the mapping to closest-feasible solutions clearer before the toy example.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The referee's summary correctly identifies the core contribution: extending recent augmented Lagrangian theory to handle infeasibility in constrained imitation learning via a hierarchical formulation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation applies external ALM results without self-reduction

full rationale

The abstract and context reference recent theoretical results on the augmented Lagrangian method in infeasible settings as the basis for the hierarchical formulation, but provide no evidence of self-citation load-bearing, fitted inputs renamed as predictions, or self-definitional reductions. The central claim maps the IL problem to a closest-feasible solution via the cited ALM properties without the derivation collapsing to its own inputs by construction. No load-bearing steps reduce to the paper's fitted parameters or prior self-citations. This is the expected non-finding for a paper whose core mapping rests on externally stated theoretical results.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Abstract-only review provides no equations, parameters, or explicit assumptions to populate the ledger; no free parameters, axioms, or invented entities can be identified.

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Cite this review

Pith. "Pith review of Infeasible optimization problems and the hierarchical augmented Lagrangian method in imitation learning." pith.science (2026). https://pith.science/paper/ZIGJJPCZ

@misc{pith2026260600730,
  author       = {Pith},
  title        = {Pith review of: Infeasible optimization problems and the hierarchical augmented Lagrangian method in imitation learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZIGJJPCZ}},
  note         = {Machine review of arXiv:2606.00730}
}

read the original abstract

Imitation learning (IL) is an effective approach to train complex robotics policies. Recent works have introduced hard constraints into imitation-learning optimization problems to ensure safety, stability, and robustness of the learned policy. However, we argue that these constraints are sometimes infeasible, which can lead to unstable or difficult training dynamics. We study a simple remedy for such situations based on recent theoretical results on the augmented Lagrangian method in infeasible settings. We show that our approach drives the learned policy toward the solution of a closest-feasible constrained IL problem with desirable properties. The method is illustrated on a toy driving example with a total-acceleration constraint and pedestrian-safety constraints, a setting in which infeasibility can naturally arise while still allowing a safe learned policy.

Figures

Figures reproduced from arXiv: 2606.00730 by the authors.

**Figure 2.** Comparison of the learned and expert total-acceleration distributions [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Control histograms for the expert and learned policies under [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 4 canonical work pages

[1]

2006 , publisher=

Numerical optimization , author=. 2006 , publisher=

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[2]

Journal of Convex Analysis , volume=

How the augmented Lagrangian algorithm can deal with an infeasible convex quadratic optimization problem , author=. Journal of Convex Analysis , volume=
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2021 American Control Conference (ACC) , pages=

On imitation learning of linear control policies: Enforcing stability and robustness constraints via LMI conditions , author=. 2021 American Control Conference (ACC) , pages=. 2021 , organization=

2021
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IEEE Control Systems Letters , volume=

Imitation learning with stability and safety guarantees , author=. IEEE Control Systems Letters , volume=. 2021 , publisher=

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Reward-Constrained Behavior Cloning. , author=. IJCAI , pages=
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arXiv preprint arXiv:2103.00452 , year=

Ekmp: Generalized imitation learning with adaptation, nonlinear hard constraints and obstacle avoidance , author=. arXiv preprint arXiv:2103.00452 , year=

work page arXiv
[7]

2022 IEEE 61st Conference on Decision and Control (CDC) , pages=

End-to-end imitation learning with safety guarantees using control barrier functions , author=. 2022 IEEE 61st Conference on Decision and Control (CDC) , pages=. 2022 , organization=

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[8]

arXiv preprint arXiv:2210.11796 , year=

Differentiable constrained imitation learning for robot motion planning and control , author=. arXiv preprint arXiv:2210.11796 , year=

work page arXiv

Show all 13 references

[9]

Mathematical Programming , volume=

The augmented Lagrangian method can approximately solve convex optimization with least constraint violation , author=. Mathematical Programming , volume=. 2023 , publisher=

2023
[10]

Proceedings of the 2024 ACM/IEEE International Conference on Human-Robot Interaction , pages=

Enhancing safety in learning from demonstration algorithms via control barrier function shielding , author=. Proceedings of the 2024 ACM/IEEE International Conference on Human-Robot Interaction , pages=

2024
[11]

arXiv preprint arXiv:2506.22428 , year=

Augmented Lagrangian methods for infeasible convex optimization problems and diverging proximal-point algorithms , author=. arXiv preprint arXiv:2506.22428 , year=
[12]

2025 IEEE International Conference on Robotics and Automation (ICRA) , pages=

Safe-gil: Safety guided imitation learning for robotic systems , author=. 2025 IEEE International Conference on Robotics and Automation (ICRA) , pages=. 2025 , organization=

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arXiv preprint arXiv:2508.03129 , year=

Safety-Aware Imitation Learning via MPC-Guided Disturbance Injection , author=. arXiv preprint arXiv:2508.03129 , year=

Pith tools

Reviewed June 28, 2026 · model on record in the stance chip above.

[1] [1]

2006 , publisher=

Numerical optimization , author=. 2006 , publisher=

2006

[2] [2]

Journal of Convex Analysis , volume=

How the augmented Lagrangian algorithm can deal with an infeasible convex quadratic optimization problem , author=. Journal of Convex Analysis , volume=

[3] [3]

2021 American Control Conference (ACC) , pages=

On imitation learning of linear control policies: Enforcing stability and robustness constraints via LMI conditions , author=. 2021 American Control Conference (ACC) , pages=. 2021 , organization=

2021

[4] [4]

IEEE Control Systems Letters , volume=

Imitation learning with stability and safety guarantees , author=. IEEE Control Systems Letters , volume=. 2021 , publisher=

2021

[5] [5]

, author=

Reward-Constrained Behavior Cloning. , author=. IJCAI , pages=

[6] [6]

arXiv preprint arXiv:2103.00452 , year=

Ekmp: Generalized imitation learning with adaptation, nonlinear hard constraints and obstacle avoidance , author=. arXiv preprint arXiv:2103.00452 , year=

work page arXiv

[7] [7]

2022 IEEE 61st Conference on Decision and Control (CDC) , pages=

End-to-end imitation learning with safety guarantees using control barrier functions , author=. 2022 IEEE 61st Conference on Decision and Control (CDC) , pages=. 2022 , organization=

2022

[8] [8]

arXiv preprint arXiv:2210.11796 , year=

Differentiable constrained imitation learning for robot motion planning and control , author=. arXiv preprint arXiv:2210.11796 , year=

work page arXiv