Primal-dual dynamical systems with closed-loop control for convex optimization in continuous and discrete time

Abstract: This paper develops a primal-dual dynamical system where the coefficients are designed in closed-loop way for solving a convex optimization problem with linear equality constraints. We first introduce a second-order primal" +first-order dual'' continuous-time dynamical system, in which both the time scaling and Hessian-driven damping are governed by a feedback control of the gradient for the Lagrangian function. This system achieves the fast convergence rates for the primal-dual gap, the feasibility violation, and the objective residual along its trajectory. Subsequently, by time discretization of this system, we develop an accelerated primal-dual algorithm with a gradient-defined adaptive step size. We also obtain convergence rates for the primal-dual gap, the feasibility violation, and the objective residual. Furthermore, we provide numerical results to demonstrate the practical efficacy and superior performance of the proposed algorithm.

• The paper introduces the first closed-loop primal-dual dynamical system with Hessian-driven damping for linearly constrained smooth convex optimization, using feedback from the Lagrangian gradient to control time scaling and damping.
• The paper proves continuous-time polynomial rates, including O(t^{-(3p+1)/2}) when q=2, and derives the adaptive AAPDA algorithm with an O(k^{-(3p-1)/(2p)}) discrete guarantee under a non-decreasing step-size condition.
• The paper reports that AAPDA outperforms several primal-dual and accelerated baselines on synthetic least-squares tasks, although trajectory convergence, general q>1 discretization, and the observed exponential behavior remain open questions.

This paper, "Primal-dual dynamical systems with closed-loop control for convex optimization in continuous and discrete time" (2602.16402), addresses the linearly equality-constrained convex optimization problem

$$\min_{x\in\mathcal{H}} f(x) \quad \text{s.t.}\quad Ax=b,$$

where $f$ is differentiable and convex with $L$-Lipschitz gradient, $A:\mathcal{H}\to\mathcal{G}$ is a continuous linear operator, and $\mathcal{S}$ (the saddle-point set of the Lagrangian) is assumed nonempty. The authors construct a "second-order primal" + "first-order dual" dynamical system in which both the time scaling and Hessian-driven damping coefficients are governed by a closed-loop feedback of the Lagrangian gradient. This is stated to be the first study of primal-dual dynamical systems with closed-loop control for this problem class, in both continuous and discrete time.

The proposed system combines an inertial primal equation with Hessian-driven damping on the Lagrangian gradient, a first-order dual ascent step evaluated at a forward extrapolated point $x(t)+\frac{\tau(t)}{\dot{\tau}(t)}\dot{x}(t)$, and two algebraic closure equations:

$$[\mu(t)]^p\left\|\nabla_x\mathcal{L}(x(t),\lambda(t))\right\|^{p-1}=1, \qquad \tau(t)=\frac{1}{q^q}\left(t_0+\int_{t_0}^t [\mu(s)]^{1/q}\,ds\right)^q,$$

with parameters $q>0$, $p\geq 1$. Because all damping coefficients are functions of the evolving trajectory through $\nabla_x\mathcal{L}(x(t),\lambda(t))$, the system is autonomous — a practical advantage over non-autonomous designs that depend explicitly on $f$0. The construction extends two lines of prior work: closed-loop controlled second-order dynamics for unconstrained problems (Lin–Jordan [lin2022]; Attouch et al. [attouchbot2025]) and mixed-order primal-dual systems with open-loop time scaling (He et al. [hehufang2022], [hetian2025]). Neither line had been combined with the other for constrained problems before this paper.

Using a Lyapunov energy function of the form

$f$1

the authors show that $f$2, yielding monotone decay of $f$3. Under $f$4 and $f$5, and via a summability lemma relating integrals of $f$6 to polynomial lower bounds on $f$7, they establish that along trajectories, for any saddle point $f$8,

$f$9

together with square-summability of $L$0 and boundedness of $L$1 with bounds depending only on initial conditions.

Two special cases sharpen the picture. For $L$2, the rate improves to $L$3, which exactly matches Lin and Jordan's rate for the unconstrained closed-loop system [lin2022] — so the extension to constrained problems costs nothing in rate. For $L$4, the rate is $L$5; when additionally $L$6 the system degenerates to an open-loop system whose rates coincide with those obtainable from the framework of He et al. [hetian2025]. Compared with Attouch et al.'s unconstrained result [attouchbot2025], which gives only $L$7, the rates here are faster, though the comparison is across different problem classes.

Restricting to $L$8 (so $L$9), an explicit-in-$A:\mathcal{H}\to\mathcal{G}$0 discretization yields AAPDA, an accelerated autonomous primal-dual algorithm. Each iteration computes the adaptive step size $A:\mathcal{H}\to\mathcal{G}$1 from the current gradient norm — no tuning parameter depending on unknown problem constants is required, which the authors argue makes implementation substantially easier than the numerical schemes derived from [lin2022]. The primal update is a proximal-type minimization combining a quadratic term around the extrapolated point $A:\mathcal{H}\to\mathcal{G}$2 and a linearized penalty $A:\mathcal{H}\to\mathcal{G}$3, followed by dual update $A:\mathcal{H}\to\mathcal{G}$4.

With a discrete Lyapunov sequence $A:\mathcal{H}\to\mathcal{G}$5 involving the corrected momentum variable $A:\mathcal{H}\to\mathcal{G}$6, the authors prove $A:\mathcal{H}\to\mathcal{G}$7 and derive, assuming $A:\mathcal{H}\to\mathcal{G}$8 is non-decreasing with $A:\mathcal{H}\to\mathcal{G}$9,

$\mathcal{S}$0

This rate approaches $\mathcal{S}$1 as $\mathcal{S}$2, and the summability condition mirrors its continuous counterpart. A caveat worth noting: the rate guarantee requires the non-decreasing, unit-bounded assumption on $\mathcal{S}$3, which is a property of the generated iterates rather than something imposed by design; the analysis does not establish that this holds in general.

Two MATLAB experiments support the theory. On a Tikhonov-regularized least-norm problem ($\mathcal{S}$4 with sparse Gaussian-generated ground truth, dimensions $\mathcal{S}$5; stopping tolerance $\mathcal{S}$6), AAPDA is compared against FPDA [hehu2022] and AALM [xu2017]. AAPDA outperforms both methods at all three scales, and — notably — the observed convergence becomes exponential as $\mathcal{S}$7 increases ($\mathcal{S}$8), far exceeding the polynomial worst-case guarantees. The Hessian-driven damping also visibly attenuates oscillations relative to FPDA. On non-negative least squares ($\mathcal{S}$9, $x(t)+\frac{\tau(t)}{\dot{\tau}(t)}\dot{x}(t)$0, densities $x(t)+\frac{\tau(t)}{\dot{\tau}(t)}\dot{x}(t)$1 and $x(t)+\frac{\tau(t)}{\dot{\tau}(t)}\dot{x}(t)$2, tolerances down to $x(t)+\frac{\tau(t)}{\dot{\tau}(t)}\dot{x}(t)$3), AAPDA is compared against FISTA [beck2009], PIA [attouchbot2025], and AFBM [a2016]: it converges fastest, reaches higher final accuracy, and its performance is stable in $x(t)+\frac{\tau(t)}{\dot{\tau}(t)}\dot{x}(t)$4, whereas the baselines' speed and accuracy degrade noticeably as the tolerance tightens. These are empirical observations on synthetic data only; no real-world datasets are tested, and the gap between the proven $x(t)+\frac{\tau(t)}{\dot{\tau}(t)}\dot{x}(t)$5 rates and the observed exponential behavior is not explained theoretically.

The paper concedes one substantive open problem directly: weak/asymptotic convergence of the solution trajectories themselves is not established for the closed-loop primal-dual system. The Lyapunov argument delivers rates and boundedness but not convergence of $x(t)+\frac{\tau(t)}{\dot{\tau}(t)}\dot{x}(t)$6 to a saddle point; the authors state that achieving such trajectory convergence likely requires redesigning both the system and the energy function. Additional limitations implicit in the results: the discrete rate theorem depends on the unverified monotonicity/lower-bound assumption on $x(t)+\frac{\tau(t)}{\dot{\tau}(t)}\dot{x}(t)$7; the algorithmic development covers only the case $x(t)+\frac{\tau(t)}{\dot{\tau}(t)}\dot{x}(t)$8, leaving the general $x(t)+\frac{\tau(t)}{\dot{\tau}(t)}\dot{x}(t)$9 discretization open; and the strong damping structure requires differentiability of $$[\mu(t)]^p\left\|\nabla_x\mathcal{L}(x(t),\lambda(t))\right\|^{p-1}=1, \qquad \tau(t)=\frac{1}{q^q}\left(t_0+\int_{t_0}^t [\mu(s)]^{1/q}\,ds\right)^q,$$0 throughout, excluding nonsmooth composite objectives except in the specific least-squares experiment where smoothness is used implicitly.

The paper contributes the first primal-dual dynamical system with closed-loop control and Hessian-driven damping for linearly constrained convex optimization, with matching continuous-time rates of order $$[\mu(t)]^p\left\|\nabla_x\mathcal{L}(x(t),\lambda(t))\right\|^{p-1}=1, \qquad \tau(t)=\frac{1}{q^q}\left(t_0+\int_{t_0}^t [\mu(s)]^{1/q}\,ds\right)^q,$$1 (at $$[\mu(t)]^p\left\|\nabla_x\mathcal{L}(x(t),\lambda(t))\right\|^{p-1}=1, \qquad \tau(t)=\frac{1}{q^q}\left(t_0+\int_{t_0}^t [\mu(s)]^{1/q}\,ds\right)^q,$$2) that recover the best known unconstrained closed-loop rates, and a practical autonomous discrete counterpart, AAPDA, achieving $$[\mu(t)]^p\left\|\nabla_x\mathcal{L}(x(t),\lambda(t))\right\|^{p-1}=1, \qquad \tau(t)=\frac{1}{q^q}\left(t_0+\int_{t_0}^t [\mu(s)]^{1/q}\,ds\right)^q,$$3 with a fully adaptive step size. Numerically, the method dominates FPDA, AALM, FISTA, PIA, and AFBM on synthetic problems, with empirically exponential behavior for large $$[\mu(t)]^p\left\|\nabla_x\mathcal{L}(x(t),\lambda(t))\right\|^{p-1}=1, \qquad \tau(t)=\frac{1}{q^q}\left(t_0+\int_{t_0}^t [\mu(s)]^{1/q}\,ds\right)^q,$$4. The main unresolved questions are trajectory convergence for the closed-loop primal-dual system and theoretical justification for the observed super-polynomial discrete convergence.