Title: Constrained Optimization From a Control Perspective via Feedback Linearization
Venue: NeurIPS 2025
Accepted: yes
Keywords: Constrained Optimization; Control Theory; Feedback Linearization
Abstract: Tools from control and dynamical systems have proven valuable for analyzing and developing optimization methods. In this paper, we establish rigorous theoretical foundations for using feedback linearization—a well-established nonlinear control technique—to solve constrained optimization problems. For equality-constrained optimization, we establish global convergence rates to first-order Karush-Kuhn-Tucker (KKT) points and uncover the close connection between the FL method and the Sequential Quadratic Programming (SQP) algorithm. Building on this relationship, we extend the FL approach to handle inequality-constrained problems. Furthermore, we introduce a momentum-accelerated feedback linearization algorithm and provide a rigorous convergence guarantee.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a method for solving constrained optimization problems using feedback linearization (FL), a concept from control theory. The authors establish global convergence rates for this method in equality-constrained problems and reveal a strong connection between the FL approach and the SQP algorithm. They then extend the method to handle inequality constraints and propose a momentum-accelerated FL algorithm that demonstrates faster convergence in practice. The theoretical claims are supported by proofs in the appendices and validated through numerical simulations on a logistic regression problem with fairness constraints.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\nThis is a solid theoretical paper that bridges the fields of nonlinear control theory and optimization. The most interesting contribution is the link established between feedback linearization and SQP, which provides a fresh perspective on a classic optimization algorithm and lends credibility to the FL approach. The development of a momentum-accelerated version is a practical and novel addition. \n\nWeaknesses:\n\nDespite its strengths, the paper has some limitations, as summarized below. Also, there is a missing 'n' in the paper title, and there might be more typos and mistakes in the main paper.\n\nQUESTIONS:\n1. In nonlinear control, the feedback linearization control law may not be globally valid. How's this issue resolved?\n\n2. The feedback linearization process often involves complex coordinate transformations and deriving the feedback control law. What is the computational complexity of the FL implementation?\n\n3. How does this approach compare against other solvers? What are potential applications?\n\nLIMITATIONS:\nThe author(s) haven't discussed their limitations in the conclusion section.\n\n1. The theoretical guarantees rely on strong conditions, such as the global boundedness of the objective and constraint function gradients (Assumption 1) and the universal invertibility of the $J_h(x)J_h(x)^\\top$ matrix (Assumption 3). These assumptions may not hold in all practical scenarios.\n\n2. While the FL-momentum method shows empirical speed-up, the theoretical analysis does not formally prove an accelerated convergence rate for the general nonconvex case; it provides the same $O(1/\\sqrt{T})$ rate as the non-accelerated method.\n\n3. The method's reliance on inverting an $m \\times m$ matrix, where $m$ is the number of constraints, could be a bottleneck for problems with a very large number of constraints.\n\n4. The analysis guarantees convergence for the best-found solution over time, which is a weaker guarantee than ensuring the final solution is optimal.\n\nI'd like to change my rating if the author(s) would like to comment on the limitations above,", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

comments: SUMMARY:
This paper builds upon the idea proposed in [15], which uses feedback linearization as a continuous analog to solve constrained optimization problems. The authors establish a sublinear convergence rate for both equality- and inequality-constrained cases. They also draw a connection between the feedback linearization framework and sequential quadratic programming (SQP), showing that an Euler discretization of the feedback linearization setup is equivalent to SQP.

STRENGTHS AND WEAKNESSES:
Strengths:
- The convergence results contribute meaningfully to the existing literature on continuous-time optimization methods.
- The connection between feedback linearization and SQP is insightful. The design of the linear dynamics sheds light on the behavior of different SQP variants with different quadratic regularization terms.
- The inclusion of a momentum-accelerated version of the algorithm, along with a partial convergence result, is a nice addition.

Weaknesses:
- The relevance of this work to the NeurIPS audience is unclear. While the paper is well-suited for optimization, control, or applied mathematics venues, its connection to machine learning is limited.
- The momentum-accelerated variant has the same convergence rate as the original algorithm and performs worse empirically than the FL-Newton variant, making its practical value unclear.
- The paper presents only one numerical example, which limits the empirical validation of the approach.

QUESTIONS:
What are some concrete application scenarios where this method would be particularly useful?

LIMITATIONS:
Yes.
TITLE: Official Review
CLARITY: 4
IMPACT: 2
ORIGINALITY: 3
DATE: 2025-07-02

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper discusses methods for solving non-convex constrained optimization using a feedback linearization approach. Specifically, the optimization process is considered as a dynamical system, where the vector field is the first derivative of the Lagrangian, and the Lagrange multipliers are inputs to the system.   The feedback linearization approach allows the authors to formulate the control law in terms of the gradients and Jacobian matrix of the objective and constraints. Then, the stability of the closed-loop system implies convergence of the states to an equilibrium point with respect to the KKT-gap. Finally, discretization of the continuous-time system shows the equivalence of the feedback linearization approach to SQP algorithms. The second part of the paper generalizes the results to inequality constrained problems.\n\nSTRENGTHS AND WEAKNESSES:\nIn general, the paper is well-structured and the contribution is sufficient. However, there are several comments.\n\nFirst, the global  convergence terminology for the paper is unusual. The authors consider the KKT-gap as a convergence metric, which should be addressed more carefully. For example, Theorems 1 and 4 show that the KKT gap goes to zero. However, the first-order KKT conditions (the KKT gap is equal $0$) are a necessary condition for optimality, so the equilibrium point $x*$ may not be a solution to the optimization problem.  Furthermore, the convergence of $x(t)$ to any equilibrium point is not guaranteed.  Thus, I would recommend adding a definition of global convergence as used in the theorems. Also, for clarity of the results, the paper should discuss the relationship between the global convergence and the convergence of $x(t)$ to the optimal solution $x*$.\n\nSecond, Lyapunov functions are required to be positive-definite everywhere except the equilibrium point, where the function must be equal $0$. However, the proposed Lyapunov functions, $l$, in theorems (Thm. 1 and 4) do not satisfy the condition. These conditions, along with non-increasing with respect to time of the Lyapunov function, guarantee the stability property of the system. Authors should avoid calling this function a Lyapunov function or modify it to satisfy the conditions.\n\nThird,  the relationship with SQP methods is interesting. However, the assumptions required for Thms. 2 and 3 are missing. For example, Thm. 3 leverages strong duality. However, the optimization in Eqn. (15) is not necessarily convex, because of non-convexity of $f$ and $h$.\n\nFinally, there are minor formatting issues. Specifically, equations in Property 3 in Theorems 3 and 4 should be formatted more accurately. For example, Theorem 3 the norm symbol in the equation is missing. The equation in Theorem~4 (Property 3) is hard to understand without additional spaces between symbols.\n\nQUESTIONS:\nI would have liked to see an analysis of how the proposed approach compares with traditional methods. The results are interesting from a theoretical perspective, but is this a better approach in general and will this line of research ultimately have a significant practical benefit?\n\nLIMITATIONS:\nno limitations acknowledged.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}