Constrained Optimization From a Control Perspective via Feedback Linearization

We thank the reviewer for the constructive feedback. We have incorporated the suggestions into the revised manuscript (see https://anonfile.io/f/wvtpAhSf). We summarize and address the reviewer’s concerns as follows:

1. Feedback linearization perspective: How does the feedback linearization viewpoint compare to existing methods, such as those in [1] and [2] cited by the reviewer?

Response: We thank the reviewer for highlighting the question. For this part, we would like to refer the reviewer to the Response to Reviewer 1 (Concern 1) who raised the same concern, as well as the Response to Concern 5 below.

The reviewer also notes that our analysis focuses on the feedback linearization of y=h(x(t)), which only captures a subset of the dynamics. This is precisely why establishing global convergence guarantees is crucial: while feedback linearization ensures constraint satisfaction, it does not in itself imply stability or convergence; thus, developing convergence guarantees serves as one of the key contributions of our paper.

2. Continuous vs. discrete-time analysis: The reviewer questions the value of continuous-time rates and whether the feedback linearization perspective can facilitate discrete-time analysis.

Response: We agree that continuous-time analysis has limitations as the reviewer suggested. However, even in continuous time, analyzing the dynamics in nonconvex settings is challenging and meaningful in its own right. Our continuous-time framework serves two purposes: (i) it offers theoretical insight into stability and convergence, and (ii) the discretization of the continuous-time dynamics directly informs discrete-time design. In this paper, we show that forward Euler discretization of the feedback-linearized dynamics yields an update closely related to SQP algorithm in discrete time. Moving forward, we believe it would be highly valuable to further explore the discrete time convergence rate and how different discretization schemes (e.g., higher-order methods) affect convergence both in theory and in practice.

3. Assumption justification: Clarification is requested regarding the necessity of the assumptions, particularly the uniqueness of the dual solution and Assumption 2.1, and its connection to constraint qualification.

Response: We thank the reviewer for raising this point. Our analysis relies on Assumption 2.1, which requires the existence of a bounded (not necessarily unique) dual variable; the uniqueness mentioned after Equation (11) is used solely for the ease of demonstration and does not affect the proofs. We have revised the manuscript to clarify our assumptions. Regarding the connection to constraint qualifications: we realize that Assumption 2.1 is closely connected to the MFCQ assumption in [3,5], which is weaker than LICQ, we have revised our discussion of the assumption (highlighted in blue).

4. Acceleration and rate focus: Why does the paper focus on the slower $1/\sqrt{T}$ convergence rate in a nonconvex setting, rather than sharper rates (e.g., $1/T$, $1/T^2$) in convex settings? How does this relate to acceleration?

Response: We would like to refer the reviewer to the Response to Reviewer 1 (Concern 2) who raised the same concern.

5. Novelty of the accelerated methods: The reviewer suggests revisiting the claim of novelty in light of recent related works, including [2,3,4,5].

Response: We sincerely thank the reviewer for pointing us to relevant works! Note that [4] primarily focuses on the setting where the constraint is an orthogonal manifold, so the setting is different from ours. While the algorithms in [2,3,5] are admittedly similar in spirit to ours, we would like to highlight a few key differences that underscore our contributions.

• In the nonconvex setting, [2,3,5] provide asymptotic convergence guarantees, whereas our work establishes non-asymptotic convergence rates. In particular, our analysis is not a trivial extension. In particular, we construct a Lyapunov function tailored to the dynamics, which is not considered in [2,3].
• We establish a connection of the method to the classical SQP algorithm, a perspective not explored in [2,3,5].
• While [5] and our momentum method share some similarities, both the continuous-time dynamics and the discretization approach differ, resulting in distinct algorithms. In the revised paper, we have updated our claims, acknowledged [5], and added a detailed comparison highlighting both the differences and similarities.

Furthermore, we believe feedback linearization provides a more general framework beyond specifying a linear target of the form \dot{y} = -Ky​. In principle, this approach allows us to leverage arbitrary stable controllers, such as PI controllers for constraint enforcement, potentially offering new algorithmic behaviors or robustness benefits. We have added corresponding additional results in Remark 2 and Appendix B.2.

We sincerely thank the reviewer for the comments and suggestions. We have revised our paper accordingly (see https://anonfile.io/f/wvtpAhSf). Below, we briefly summarize the reviewer's concerns and then address them one by one:

1. Lack of Clarity in Assumptions and Paper Structure: The reviewer noted that key assumptions are not clearly or consistently stated, and that convergence criteria (KKT gap) is given in an elaborative manner.

Response: We thank the reviewer for the feedback. Our intention was to present a streamlined development by stating assumptions only when they are required for specific theoretical results, rather than listing all assumptions upfront. This reflects the fact that the algorithm itself applies broadly to differentiable objective and constraint functions, and that many of the stronger assumptions (e.g., LICQ, smoothness) are only invoked for particular convergence guarantees. That said, we recognize that this design choice may have caused confusion. To improve clarity, we will add a brief clarification at the beginning of the theoretical section stating that all functions are assumed to be differentiable, and that additional assumptions will be introduced where needed to support the analysis.

To further address the reviewer’s confusion in the convergence criteria, we have moved the definition of KKT-gap earlier into the problem formulation part rather than right before stating the theorem.

2. Restrictive Assumptions: The reviewer found the assumptions underlying the theoretical analysis—such as LICQ and full row-rank Jacobians—too strong or uncommon in practice, and requested better justification and examples where they hold.

Response: We thank the reviewer for raising this concern. We emphasize that the assumptions invoked in our analysis—such as LICQ (Assumption 1.1)—are standard in the literature on constrained optimization and SQP (see Appendix A paragraph labeed “SQP”). Further, for the inequality-constrained setting, our main convergence analysis relies on Assumption 2.1, which is strictly weaker than LICQ; we clarify this relationship in the discussion following Assumption 2.1 and formally prove it in Appendix F. We also acknowledge that these assumptions may not hold in all practical scenarios; to this end, we include a discussion in Appendix A to highlight their limitations and the potential for extensions beyond the current setting. We would also like to refer the reviewer to our response to Reviewer 3 (Concern 3), who raised a related point.

3. Experimental Setup and Justification: The reviewer questioned whether the experimental problems satisfy the theoretical assumptions, and found that the connection between the experiments and the theory was not sufficiently clear.

Response: We thank the reviewer for this helpful observation. We clarify that the models used in our experiments (including Heterogenous Logistic Regression and ACOPF) are constructed to satisfy the standing assumptions in our analysis. In the revised paper, we explicitly state which assumptions are met in each experimental setting.

At the same time, we would like to emphasize that it is often valuable for numerical experiments to test methods beyond the strict confines of theoretical assumptions, as strong empirical performance can indicate the method’s robustness in practice. And indeed, one of the strengths of our approach is that it can be applied without explicitly checking strong conditions.

4. Relation to SQP and Novelty: The reviewer felt that the contribution appears closely tied to prior work such as Cerone et al. (2024) and was not fully convinced of the added value or novelty of the proposed approach over established SQP methods.

Response: We thank the reviewer for raising this point. We would like to emphasize several aspects of our contribution that we believe are novel and meaningful.

First, in contrast to prior works such as Cerone et al. that provide only local stability guarantees in the nonconvex setting, our analysis establishes non-asymptotic global convergence rates. This represents a nontrivial advancement, particularly given the complexity of constrained, nonconvex dynamics.

Secondly, compared with the SQP algorithm, we proposed a novel momentum algorithm, which is different from classical SQP formulations and is able to accelerate convergence in using only first-order information. Additionally, compared with the SQP algorithm, the control-theoretic lens we adopt—particularly through feedback linearization—offers a broader design framework. While our current work targets a specific dynamic, this viewpoint allows the incorporation of more general stabilizing controllers for constraint satisfaction, potentially leading to new classes of algorithms with desirable robustness or geometric properties. We would also like to point the reviewer to our response to Reviewer 3 (Concern 5), who raised a related point.

We sincerely thank the reviewer for carefully reading our paper and providing constructive feedback, as well as for their positive assessment of our work and its positioning within the broader literature on optimization and control. We are also grateful for the reviewer’s detailed suggestions on typos and improvements to the phrasing of theorems and statements, which have helped us enhance the clarity of the manuscript. With regard to the reviewer’s comments on typos or notational inconsistencies, if we did not provide a specific response, it means we fully agree with the reviewer’s observation and have corrected the issue accordingly in the revised manuscript (see https://anonfile.io/f/DEcRCSxS). Below, we address the technical questions raised by the reviewer.

1. Elaboration on FL-momentum: "line 325 on the right: what does “Note that the only difference in (16) is that we add a momentum step w_t = x_t +\beta(x_t −x_{t−1} )” - - - mean? The only difference as opposed to what?"

Response: We sincerely apologize for the confusion. In this part, we intended to compare Equation (16) (now Equation (18) in the revised version) with Equation (6) (now Equation (8)), where FL-proximal is used without momentum. We have revised the text for clarity as follows: “Note that compared with FL-proximal (8), the difference in (18) is the addition of a momentum step $w_t = x_t + \beta(x_t - x_{t-1})$.”

We hope this revision helps clarify the point, and we truly appreciate the reviewer’s feedback. We would be happy to further discuss this if there are any remaining questions.

2. Question about Figure 3: “Isn’t it weird that the blue line on the right of Figure 3 is missing completely? The authors claim that it “nearly overlaps with the FL-momentum curve”, but it seems improbable that we can’t see it at all. ”

Response: We sincerely thank the reviewer for carefully examining the details of our numerical experiments. In the IEEE 118-bus example, we observed that the algorithm becomes unstable when the momentum hyperparameter $\beta$ is set too large. The best-performing value found during hyperparameter tuning was $\beta = 0.02$, which results in the FL-proximal and FL-momentum curves appearing nearly identical. We believe this is due to the ill-conditioned optimization landscape of the problem, which limits the benefit of momentum. To clarify this, we have plotted a zoomed-in version of the learning curves (see here https://anonfile.io/f/huBzM42f), where subtle differences between the two methods can be observed. We greatly appreciate the reviewer’s observation and have double-checked the plot to ensure its accuracy.

We sincerely thank the reviewer for the comments and suggestions! We have revised our paper (see https://anonfile.io/f/DEcRCSxS) accordingly. We briefly summarize the reviewer’s main concerns as follows and address them one by one:

1. Missing Discussion on First-Order Methods: The reviewer suggested that the paper should include a more explicit comparison with the convergence rates of standard first-order methods, such as primal-dual gradient descent (PDGD), projected gradient descent (PGD), and augmented Lagrangian methods (ALM).

Response: We thank the reviewer for raising this point. We would like to point the reviewer to Appendix A for a detailed comparison with existing first order methods, which we briefly summarize as follows:

PDGD has been extensively studied (e.g., Kose, 1956; Wang and Elia, 2011; Qu and Li, 2018), but is primarily restricted to convex settings. Several works suggest that PDGD can fail to converge or behave poorly in nonconvex problems. In contrast, our feedback linearization framework remains applicable in nonconvex settings, as supported by our theory and experiments. PGD also becomes unsuitable in nonconvex problems, as projections onto general nonconvex sets are often computationally intractable.

While ALM can handle nonconvex constraints, each iteration involves solving a nonconvex subproblem, which may be costly. Moreover, in settings where the constraint dimension is much smaller than the dimension of 𝑥, SQP-based approaches (such as ours) tend to perform better numerically.

In the revised paper, we have added corresponding numerical results to compare these methods in Appendix B.1. We thank the reviewer for helping improve the clarity and reliability of our presentation.

2. Claim About Acceleration and Convergence Rate: The reviewer found the abstract potentially misleading, as it mentions acceleration without demonstrating an improved convergence rate over the baseline method.

Response: We thank the reviewer for this helpful observation and apologize for the ambiguity in the current phrasing. As noted in our response to Concern 1, a key strength of the feedback linearization (FL) framework is its applicability to nonconvex problems. In such settings, even momentum-accelerated methods are only known to achieve a convergence rate of $O(1/\sqrt{T})$, which is the rate we aim to establish in this work.

Although convex optimization is not the focus of the paper, we note that our accelerated algorithm can be connected to momentum-accelerated projected gradient descent in the convex setting (see Appendix E.1). This connection allows us to leverage existing convergence results for projected accelerated methods, providing theoretical justification for the observed acceleration of our algorithm in the convex regime. We have revised the claim in our paper to clarify the confusion (Remark 5).

3. Second-Order Nature and Scalability: The reviewer noted that the algorithm requires matrix inversion, placing it closer to second-order methods, and raised concerns about its scalability to high-dimensional problems.

Response: We thank the reviewer for raising this point. First, we would like to clarify that although our methods (FL-proximal and FL-momentum) involve matrix inversion, they are not second-order methods, as they only require first-order information—specifically, the gradient of $f$ and the Jacobian of $h$; no Hessians or second-order derivatives are used.

Second, regarding scalability: the matrix inversion arises in a subspace defined by the constraints. In many practical settings, e.g. safe RL, the number of constraints is significantly smaller than the dimension of $x$. In such cases, the inversion is computationally inexpensive and can be performed efficiently. We will clarify this in the revision and emphasize that the cost of inversion depends on the number of constraints, not the ambient dimension.

We appreciate the reviewer’s comment on scalability, and we have incorporated the above discussion in the revised manuscript (Remark 1).

4. Lack of Computational Complexity Analysis

Response: We thank the reviewer for this observation and have added Remark 1 to discuss computational complexity in the revised manuscript.

5. Convergence Rate Dependencies: The reviewer observed that the convergence rates established in Theorems 1 and 4 do not explicitly characterize dependencies on key problem parameters, such as dimensionality, number of constraints, or condition numbers.

Response: We appreciate the reviewer’s comment on this point. The convergence rates depend not directly on the dimensionality or number of constraints, but on parameters such as M and D (from Assumption 1), and the condition number of the matrix T(x). (See e.g. the detailed convergence rate formula in Statement 4 within Theorem 1 and 4).

Once again, we appreciate the reviewer’s feedback and are open to further discussions.