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arxiv: 2604.06511 · v2 · submitted 2026-04-07 · 🧮 math.OC · cs.SY· eess.SY

Feedback control of Lagrange multipliers for non-smooth constrained optimization

V. Cerone , S. M. Fosson , S. Pirrera , A. Re , D. Regruto This is my paper

Pith reviewed 2026-05-10 18:24 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords non-smooth optimizationconstrained optimizationaugmented LagrangianLagrange multipliersfeedback controlexponential convergenceproximal methodsdynamical systems
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The pith

A control-theoretic approach using feedback on Lagrange multipliers produces two new algorithms for non-smooth constrained optimization with global exponential convergence.

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

The paper develops a framework that models non-smooth constrained optimization as a control problem by treating the proximal augmented Lagrangian as a dynamical plant. Static and dynamic feedback controllers are then designed to act on the Lagrange multipliers and steer the system to the desired equilibria. Both resulting algorithms are proven to converge globally and exponentially to the solution when the objective function is strongly convex. This provides a systematic way to derive and certify optimization methods that handle non-differentiable terms, which appear in applications such as sparse signal recovery and image reconstruction.

Core claim

By representing the proximal augmented Lagrangian as a plant whose equilibria correspond to the stationary points of the original problem, the authors apply static and dynamic feedback controllers to the multipliers. This produces two closed-loop systems that converge globally and exponentially to those equilibria under strong convexity assumptions.

What carries the argument

The dynamical plant obtained from the proximal augmented Lagrangian, with feedback applied directly to the Lagrange multipliers.

Load-bearing premise

The proximal augmented Lagrangian formulation can be accurately modeled as a dynamical plant whose equilibria coincide with the stationary points of the non-smooth constrained problem.

What would settle it

A strongly convex non-smooth constrained problem together with a simulation or calculation showing that the trajectories of either closed-loop system fail to reach the known optimum at an exponential rate.

Figures

Figures reproduced from arXiv: 2604.06511 by A. Re, D. Regruto, S. M. Fosson, S. Pirrera, V. Cerone.

Figure 1
Figure 1. Figure 1: Scheme of the CMO approach: the dynamical system [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Residual ∥Ax − y∥2 versus ∥x∥1 averaged over 100 runs. Compar￾ison between the proposed dynamic Prox-CMO, I-ISTA [35], PI-PGD [25], AD-ISTA [36], and the gradient descent method. All the results are averaged over 100 random runs [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Evolution of the residual of the CT algorithms [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of the sparsity level ∥x(k)∥0 averaged over 100 runs. 6.2 Shidoku puzzle The second numerical example is a problem with a nonsmooth cost function and non-convex polynomial constraints. Shidoku is a 4x4 version of the 9x9 Sudoku puzzle. Given an initial scheme as the one reported in [PITH_FULL_IMAGE:figures/full_fig_p027_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Evolution of the support error Pn i=1 |ι(xi(k)) − ι(xei)| averaged over 100 runs. 1 4 2 3 (a) Initial scheme 3 1 2 4 4 2 1 3 2 4 3 1 1 3 4 2 (b) Solved scheme [PITH_FULL_IMAGE:figures/full_fig_p028_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Shidoku puzzle and its solution. Bold entries indicate the original [PITH_FULL_IMAGE:figures/full_fig_p028_6.png] view at source ↗
read the original abstract

In this work, we develop a control-theoretic framework for constrained optimization problems with composite objective functions including non-differentiable terms. Building on the proximal augmented Lagrangian formulation, we construct a plant whose equilibria correspond to the stationary points of the optimization problem. Within this framework, we propose two control strategies - a static controller and a dynamic controller - leading to two novel optimization algorithms. We provide a theoretical analysis, establishing global exponential convergence under strong convexity assumptions. Finally, we demonstrate the effectiveness of the proposed methods through numerical experiments, benchmarking their performance against state-of-the-art approaches.

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.

Referee Report

1 major / 2 minor

Summary. The paper develops a control-theoretic framework for non-smooth constrained optimization problems with composite objectives. It models the proximal augmented Lagrangian as a dynamical plant whose equilibria are asserted to coincide with the stationary points of the original problem, then designs static and dynamic feedback controllers on the Lagrange multipliers to produce two new optimization algorithms. The central theoretical result is a proof of global exponential convergence under strong convexity assumptions, accompanied by numerical benchmarking against existing methods.

Significance. If the plant-equilibria correspondence and the convergence analysis are rigorously established, the work offers a novel bridge between feedback control and non-smooth optimization that could inspire new algorithm designs with explicit exponential rates. The inclusion of numerical experiments benchmarking against state-of-the-art solvers is a constructive element that helps assess practical relevance.

major comments (1)
  1. The central claim that the equilibria of the proximal augmented Lagrangian plant coincide exactly with the subdifferential stationarity conditions of the non-smooth constrained problem (including the proximal mapping for the non-differentiable term) is load-bearing for all subsequent convergence results. The manuscript must supply an explicit lemma or proposition (with proof) verifying this equivalence, addressing possible gaps arising from constraint qualifications or inexact proximal embeddings; without it, the global exponential convergence statements cannot be guaranteed to apply to the true optima of the original problem.
minor comments (2)
  1. Notation for the plant dynamics and controller gains should be introduced with a single consistent table or list of symbols to improve readability across sections.
  2. The numerical experiments section would benefit from reporting iteration counts or CPU times in addition to objective values to allow direct comparison of convergence speed with the claimed exponential rates.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback. The suggestion to make the equilibria correspondence explicit strengthens the manuscript, and we will incorporate a dedicated lemma with proof in the revision.

read point-by-point responses

  1. Referee: The central claim that the equilibria of the proximal augmented Lagrangian plant coincide exactly with the subdifferential stationarity conditions of the non-smooth constrained problem (including the proximal mapping for the non-differentiable term) is load-bearing for all subsequent convergence results. The manuscript must supply an explicit lemma or proposition (with proof) verifying this equivalence, addressing possible gaps arising from constraint qualifications or inexact proximal embeddings; without it, the global exponential convergence statements cannot be guaranteed to apply to the true optima of the original problem.

    Authors: We agree that an explicit verification of the equilibria-stationarity equivalence is necessary to underpin the convergence analysis. In the revised manuscript we add a new Lemma 1 (Section 2) whose proof shows that any equilibrium of the proximal augmented Lagrangian plant satisfies the subdifferential inclusion for the nonsmooth term via the proximal mapping definition, together with the stationarity conditions for the smooth objective and the equality constraints. The proof uses the optimality condition of the proximal operator and the definition of the augmented Lagrangian. We now explicitly list the standing assumption of the linear independence constraint qualification (LICQ) to guarantee well-defined Lagrange multipliers and rule out qualification gaps. Our framework assumes exact proximal evaluations, as is standard for proximal algorithms; we clarify this assumption in the revised text and note that inexact proximal mappings would require separate error-bound analysis, which lies outside the present scope. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper constructs a dynamical plant from the proximal augmented Lagrangian such that equilibria are defined to coincide with the stationary points of the non-smooth problem; this is an explicit modeling choice rather than a reduction of the convergence result to its own inputs. The static and dynamic controllers are then applied to drive the plant to those equilibria, with global exponential convergence proven under strong convexity via standard Lyapunov analysis. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain. The approach remains externally verifiable through the reported numerical benchmarks against independent state-of-the-art methods.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on treating the proximal augmented Lagrangian as a valid plant model and on strong convexity to obtain exponential convergence; no free parameters or invented entities are indicated in the abstract.

axioms (2)
  • domain assumption The objective function is strongly convex
    Invoked to guarantee global exponential convergence of the closed-loop system.
  • domain assumption Equilibria of the proximal augmented Lagrangian plant coincide with stationary points of the original problem
    Foundational modeling step stated in the abstract.

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