CMC-Opt: Constraint Manifold with Corners for Inequality-Constrained Optimization

Frank Dellaert; Yetong Zhang

arxiv: 2605.20796 · v1 · pith:JCRRCLIZnew · submitted 2026-05-20 · 💻 cs.RO

CMC-Opt: Constraint Manifold with Corners for Inequality-Constrained Optimization

Yetong Zhang , Frank Dellaert This is my paper

Pith reviewed 2026-05-21 04:51 UTC · model grok-4.3

classification 💻 cs.RO

keywords constraint manifoldsinequality constraintsmanifold optimizationkinodynamic planningrobot trajectory optimizationnonlinear constraintsfeasible space optimization

0 comments

The pith

Constraint manifolds with corners turn mixed equality and inequality constrained optimization into unconstrained manifold optimization.

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

The paper presents a manifold-based framework for optimization problems that include both equality and inequality constraints, as commonly arise in robotics. It defines constraint manifolds with corners to capture the feasible state space directly. This reformulation lets the original problem be solved as an unconstrained optimization task using extended manifold algorithms. A sympathetic reader would care because the method promises reliable, dynamically feasible solutions for large-scale planning tasks where penalty methods or other standard approaches break down.

Core claim

We introduce constraint manifolds with corners to represent the state space satisfying mixed nonlinear equality and inequality constraints. We extend manifold optimization algorithms to operate on this new topological structure, thereby transforming the original constrained optimization problem into an unconstrained one directly on the constrained state space.

What carries the argument

Constraint manifolds with corners, the topological structure that encodes equality constraints as the base manifold and inequality constraints as its corner boundaries, allowing direct application of manifold optimization without auxiliary constraint terms.

If this is right

Robotics optimization problems with mixed constraints can be solved using unmodified manifold optimization routines after the feasible space is cast as a manifold with corners.
Large-scale kinodynamic planning tasks produce dynamically feasible trajectories in cases where conventional constrained optimization methods fail.
Penalty functions and barrier methods are no longer required for handling inequality constraints within the manifold framework.
Manifold optimization gains a direct route to problems whose feasible sets are defined by both equalities and inequalities.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same corner-manifold construction could be applied in other domains such as graphics or optimal control where feasible sets are carved by inequalities.
If corner handling proves efficient, the technique may support real-time replanning loops in autonomous systems.
Library implementations could make this the default geometric solver for trajectory optimization in robotics.

Load-bearing premise

Mixed nonlinear equality and inequality constraints can be represented as a constraint manifold with corners on which existing manifold optimization algorithms can be extended without introducing new failure modes or prohibitive computational overhead.

What would settle it

A kinodynamic planning benchmark where the extended manifold optimizer on the constraint manifold with corners either fails to produce a feasible trajectory or requires more computation time than a standard penalty-based solver on the identical problem.

Figures

Figures reproduced from arXiv: 2605.20796 by Frank Dellaert, Yetong Zhang.

**Figure 2.** Figure 2: (a) Factor graph representation of a constrained optimization problem. [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Figure 3: (a) Example constraint manifolds with corners. Top is defined by [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗

**Figure 4.** Figure 4: Example of Riemannian gradient descent applied on a “half sphere” [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗

read the original abstract

We introduce a manifold-based framework for addressing optimization problems with equality and inequality constraints found in robotics. Our approach transforms the original problem into an unconstrained optimization problem directly on the constrained state space. To achieve this, we introduce ``constraint manifolds with corners" to represent the state space satisfying mixed nonlinear equality and inequality constraints. We further extend manifold optimization algorithms to operate on this new topological structure. We demonstrate the power and robustness of our framework in the context of a large-scale kinodynamic planning problem, successfully generating dynamically feasible trajectories where standard methods fail.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines constraint manifolds with corners to fold mixed nonlinear constraints into a space for direct manifold optimization, with a robotics planning demo, but the construction likely needs explicit regularity conditions at corners to avoid breakdowns.

read the letter

The main point is that they build a new structure called constraint manifolds with corners so that equality and inequality constraints together define a feasible set that looks like a manifold with half-space boundaries. This lets them run existing manifold optimization tools directly on it instead of using penalties or projections for the inequalities. The claim is that this turns the original constrained problem into an unconstrained one on that space, and they show it on a large kinodynamic planning task where standard methods could not produce feasible trajectories. That demo is the strongest part so far, because it targets a setting where feasibility really matters and current solvers struggle at scale. The construction itself appears new relative to the usual manifold optimization literature, which mostly assumes equalities only. Extending the retraction and gradient steps to handle the corners is the technical step they add. On the soft side, the stress-test concern lands: at points where several inequalities are active, the active gradients must stay linearly independent for the tangent cone to be a proper product of half-spaces and for the local charts to work. The abstract gives no sign that they state or check a condition like LICQ or MFCQ at those intersections. If that fails, the manifold structure collapses locally and the extended steps become undefined or jumpy. The full paper may address this with proofs or implementation safeguards, but without seeing that it is hard to judge whether the transformation stays valid everywhere. The work is aimed at robotics researchers who already use manifold methods for planning or control and want a cleaner way to incorporate inequalities. A reader who cares about constraint handling in nonlinear optimization would get something concrete from the planning example and the algorithmic extension. It is worth sending to peer review because the practical result is useful and the idea is worth testing in detail, even if the theoretical foundations need tightening on the regularity side.

Referee Report

2 major / 2 minor

Summary. The paper introduces 'constraint manifolds with corners' to represent feasible sets defined by mixed nonlinear equality and inequality constraints in robotics. It transforms the original constrained optimization problem into an unconstrained one on this manifold structure and extends existing manifold optimization algorithms (including retractions and Riemannian gradients) to operate on it. The framework is demonstrated on a large-scale kinodynamic planning task, where it produces dynamically feasible trajectories that standard constrained optimization methods fail to generate.

Significance. If the manifold-with-corners construction is shown to be well-defined under standard constraint qualifications and the algorithmic extensions preserve convergence properties, the work could offer a principled way to incorporate inequality constraints directly into manifold-based planning and control pipelines in robotics. The empirical demonstration on kinodynamic planning provides initial evidence of practical utility where conventional approaches break down.

major comments (2)

[§3 (Definition of CMC)] Definition of constraint manifold with corners (likely §3 or §4): The manuscript does not state or verify any regularity condition (LICQ, MFCQ, or equivalent) at corner loci where multiple inequality constraints become active. Without linear independence of the active gradients, the tangent cone is not guaranteed to be a product of half-spaces, the local chart structure fails, and the claimed retraction / Riemannian gradient steps become undefined or discontinuous. This directly undermines the central claim that the feasible set forms a manifold on which standard manifold optimizers can be extended without new failure modes.
[§5 (Algorithmic extension)] Extension of manifold optimization algorithms (likely §5): The paper asserts that existing retraction and vector transport operators can be extended to the corner structure, but provides no proof or counter-example analysis showing that the extended operators remain smooth or satisfy the necessary first-order conditions at the boundary strata. If the extension is only heuristic, the transformation to an 'unconstrained' problem on the manifold is no longer rigorously valid.

minor comments (2)

[§3] Notation for the corner strata and active-set index sets is introduced without a clear table or diagram; readers must reconstruct the indexing from prose.
[§6 (Experiments)] The kinodynamic planning experiment reports success where 'standard methods fail' but does not quantify failure rates, iteration counts, or constraint violation metrics for the baselines, making the robustness claim difficult to assess.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. The comments raise important points regarding the theoretical foundations of constraint manifolds with corners, particularly at the corner loci. We address each major comment below and indicate the revisions we will make to the manuscript.

read point-by-point responses

Referee: Definition of constraint manifold with corners: The manuscript does not state or verify any regularity condition (LICQ, MFCQ, or equivalent) at corner loci where multiple inequality constraints become active. Without linear independence of the active gradients, the tangent cone is not guaranteed to be a product of half-spaces, the local chart structure fails, and the claimed retraction / Riemannian gradient steps become undefined or discontinuous. This directly undermines the central claim that the feasible set forms a manifold on which standard manifold optimizers can be extended without new failure modes.

Authors: We agree that the regularity conditions must be explicitly stated to ensure the validity of the manifold structure at corners. In the revised manuscript, we will add to §3 a clear statement that we assume the Mangasarian-Fromovitz Constraint Qualification (MFCQ) holds at all points, including corner loci. Under MFCQ, the active gradients are positively linearly independent, guaranteeing that the tangent cone is a product of half-spaces. This preserves the local chart structure and ensures that the extended retraction and Riemannian gradient are well-defined and continuous. We will include a short proof sketch based on standard results in nonlinear programming. revision: yes
Referee: Extension of manifold optimization algorithms: The paper asserts that existing retraction and vector transport operators can be extended to the corner structure, but provides no proof or counter-example analysis showing that the extended operators remain smooth or satisfy the necessary first-order conditions at the boundary strata. If the extension is only heuristic, the transformation to an 'unconstrained' problem on the manifold is no longer rigorously valid.

Authors: We acknowledge that the manuscript lacks a formal proof for the properties of the extended operators. To address this, we will revise §5 to include a proposition establishing that, under the MFCQ assumption, the extended retraction remains continuous at the boundary strata and that the Riemannian gradient satisfies the first-order necessary conditions for optimality. We will also provide a counter-example illustrating potential issues without the qualification, thereby confirming the rigorous validity of the unconstrained formulation on the CMC. revision: yes

Circularity Check

0 steps flagged

No circularity: new representation and extension presented as independent construction

full rationale

The paper defines a novel 'constraint manifolds with corners' structure to encode mixed equality/inequality constraints and then extends standard manifold optimization operators to this structure, thereby converting the original constrained problem into an unconstrained one on the new space. No equations or derivations in the provided text reduce a claimed result to a fitted parameter, a self-citation chain, or a renamed input; the central transformation is introduced by construction of the new topological object rather than recovered from prior outputs. The kinodynamic planning demonstration functions as external validation, not as a tautological input to the framework itself. The derivation chain therefore remains self-contained against external manifold-optimization literature.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no information on free parameters, axioms, or invented entities; ledger left empty.

pith-pipeline@v0.9.0 · 5609 in / 991 out tokens · 29797 ms · 2026-05-21T04:51:22.948450+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages · 1 internal anchor

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Continuous-time gaussian pro- cess motion planning via probabilistic inference.The In- ternational Journal of Robotics Research, 37(11):1319– 1340, 2018

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Yetong Zhang, Fan Jiang, Gerry Chen, Varun Agrawal, Adam Rutkowski, and Frank Dellaert. Constraint man- ifolds for robotic inference and planning. In2023 IEEE International Conference on Robotics and Automa- tion (ICRA), pages 3175–3181, 2023. doi: 10.1109/ ICRA48891.2023.10161024

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Absil, R

P.-A. Absil, R. Mahony, and R. Sepulchre.Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton, NJ, 2008. ISBN 978-0-691-13298-3

work page 2008

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Nonlinear programming.Journal of the Operational Research Society, 48(3):334–334, 1997

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SIAM, 2010

John T Betts.Practical methods for optimal control and estimation using nonlinear programming. SIAM, 2010

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An introduction to optimization on smooth manifolds

Nicolas Boumal. An introduction to optimization on smooth manifolds. Available online, Aug 2020. URL http://www.nicolasboumal.net/book

work page 2020

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An inexact sqp method for equality constrained optimization

Richard H Byrd, Frank E Curtis, and Jorge Nocedal. An inexact sqp method for equality constrained optimization. SIAM Journal on Optimization, 19(1):351–369, 2008

work page 2008

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Ddf-sam: Fully distributed slam using constrained factor graphs

Alexander Cunningham, Manohar Paluri, and Frank Del- laert. Ddf-sam: Fully distributed slam using constrained factor graphs. In2010 IEEE/RSJ International Confer- ence on Intelligent Robots and Systems, pages 3025–

work page

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Dellaert

F. Dellaert. Factor graphs and GTSAM: A hands- on introduction. Technical report, Georgia Institute of Technology, 2012

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Dellaert and M

F. Dellaert and M. Kaess.Factor Graphs for Robot Perception. Now Publishers Inc., August 2017

work page 2017

[9] [9]

On manifolds with corners

Dominic Joyce. On manifolds with corners.arXiv preprint arXiv:0910.3518, 2009

work page internal anchor Pith review Pith/arXiv arXiv 2009

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Stomp: Stochastic trajectory optimization for motion planning

Mrinal Kalakrishnan, Sachin Chitta, Evangelos Theodorou, Peter Pastor, and Stefan Schaal. Stomp: Stochastic trajectory optimization for motion planning. In2011 IEEE international conference on robotics and automation, pages 4569–4574. IEEE, 2011

work page 2011

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F. R. Kschischang, B. J. Frey, and H. . Loeliger. Factor graphs and the sum-product algorithm.IEEE Transac- tions on Information Theory, 47(2):498–519, 2001. doi: 10.1109/18.910572

work page doi:10.1109/18.910572 2001

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Optimization- based locomotion planning, estimation, and control de- sign for the atlas humanoid robot.Autonomous robots, 40(3):429–455, 2016

Scott Kuindersma, Robin Deits, Maurice Fallon, Andr ´es Valenzuela, Hongkai Dai, Frank Permenter, Twan Koolen, Pat Marion, and Russ Tedrake. Optimization- based locomotion planning, estimation, and control de- sign for the atlas humanoid robot.Autonomous robots, 40(3):429–455, 2016

work page 2016

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Springer, 2012

John M Lee and John M Lee.Smooth manifolds. Springer, 2012

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Lynch and Frank C

Kevin M. Lynch and Frank C. Park.Modern Robotics: Mechanics, Planning, and Control. Cambridge Univer- sity Press, USA, 1st edition, 2017. ISBN 1107156300

work page 2017

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Aaron Mahler, Tyrus Berry, Tom Stephens, Harbir Antil, Michael Merritt, Jeanie Schreiber, and Ioannis Kevrekidis. On-manifold projected gradient descent. Frontiers in Computer Science, 6:1274181, 2024

work page 2024

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Crocoddyl: An efficient and versatile framework for multi-contact optimal control

Carlos Mastalli, Rohan Budhiraja, Wolfgang Merkt, Guilhem Saurel, Bilal Hammoud, Maximilien Naveau, Justin Carpentier, Ludovic Righetti, Sethu Vijayakumar, and Nicolas Mansard. Crocoddyl: An efficient and versatile framework for multi-contact optimal control. In 2020 IEEE International Conference on Robotics and Automation (ICRA), pages 2536–2542. IEEE, 2020

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Continuous-time gaussian pro- cess motion planning via probabilistic inference.The In- ternational Journal of Robotics Research, 37(11):1319– 1340, 2018

Mustafa Mukadam, Jing Dong, Xinyan Yan, Frank Del- laert, and Byron Boots. Continuous-time gaussian pro- cess motion planning via probabilistic inference.The In- ternational Journal of Robotics Research, 37(11):1319– 1340, 2018

work page 2018

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Wright.Numerical Opti- mization

Jorge Nocedal and Stephen J. Wright.Numerical Opti- mization. Springer, New York, NY , USA, second edition, 2006

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Hybrid direct colloca- tion and control in the constraint-consistent subspace for dynamic legged robot locomotion

Diego Pardo, Michael Neunert, Alexander Winkler, Ruben Grandia, and Jonas Buchli. Hybrid direct colloca- tion and control in the constraint-consistent subspace for dynamic legged robot locomotion. InRobotics: Science and Systems, 07 2017. doi: 10.15607/RSS.2017.XIII.042

work page doi:10.15607/rss.2017.xiii.042 2017

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A direct method for trajectory optimization of rigid bodies through contact.The International Journal of Robotics Research, 33(1):69–81, 2014

Michael Posa, Cecilia Cantu, and Russ Tedrake. A direct method for trajectory optimization of rigid bodies through contact.The International Journal of Robotics Research, 33(1):69–81, 2014

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Motion planning with sequential convex optimization and convex colli- sion checking.The International Journal of Robotics Research, 33(9):1251–1270, 2014

John Schulman, Yan Duan, Jonathan Ho, Alex Lee, Ibrahim Awwal, Henry Bradlow, Jia Pan, Sachin Patil, Ken Goldberg, and Pieter Abbeel. Motion planning with sequential convex optimization and convex colli- sion checking.The International Journal of Robotics Research, 33(9):1251–1270, 2014

work page 2014

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A review on bilevel optimization: From classical to evolu- tionary approaches and applications.IEEE transactions on evolutionary computation, 22(2):276–295, 2017

Ankur Sinha, Pekka Malo, and Kalyanmoy Deb. A review on bilevel optimization: From classical to evolu- tionary approaches and applications.IEEE transactions on evolutionary computation, 22(2):276–295, 2017

work page 2017

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A factor graph approach to estimation and model pre- dictive control on unmanned aerial vehicles

Duy-Nguyen Ta, Marin Kobilarov, and Frank Dellaert. A factor graph approach to estimation and model pre- dictive control on unmanned aerial vehicles. In2014 International Conference on Unmanned Aircraft Systems (ICUAS), pages 181–188. IEEE, 2014

work page 2014

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Constrained nonlinear least squares

Lieven Vandenberghe. Constrained nonlinear least squares. http://www.seas.ucla.edu/ ∼vandenbe/133B/ lectures/nllseq.pdf, 2020

work page 2020

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A factor-graph approach for optimization problems with dynamics constraints.arXiv preprint arXiv:2011.06194, 2020

Mandy Xie, Alejandro Escontrela, and Frank Dellaert. A factor-graph approach for optimization problems with dynamics constraints.arXiv preprint arXiv:2011.06194, 2020

work page arXiv 2011

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Grounding language with visual affordances over un- structured data

Yetong Zhang, Fan Jiang, Gerry Chen, Varun Agrawal, Adam Rutkowski, and Frank Dellaert. Constraint man- ifolds for robotic inference and planning. In2023 IEEE International Conference on Robotics and Automa- tion (ICRA), pages 3175–3181, 2023. doi: 10.1109/ ICRA48891.2023.10161024

work page arXiv 2023

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Chomp: Covariant hamiltonian optimization for motion planning

Matt Zucker, Nathan Ratliff, Anca D Dragan, Mihail Piv- toraiko, Matthew Klingensmith, Christopher M Dellin, J Andrew Bagnell, and Siddhartha S Srinivasa. Chomp: Covariant hamiltonian optimization for motion planning. The International journal of robotics research, 32(9-10): 1164–1193, 2013

work page 2013