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

Huan Zhang, Xiangkai Sun, Shengjie Li, Kok Lay Teo

TL;DR

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 and develops an accelerated primal-dual algorithm with a gradient-defined adaptive step size.

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.

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

TL;DR

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 and develops an accelerated primal-dual algorithm with a gradient-defined adaptive step size.

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.
Paper Structure (10 sections, 7 theorems, 104 equations, 1 algorithm)

This paper contains 10 sections, 7 theorems, 104 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem description
  3. Motivation and related works
  4. Dynamical systems with Hessian-driven damping
  5. Dynamical systems with closed-loop control
  6. Main contributions
  7. Organization
  8. The continuous-time dynamical system
  9. The discrete accelerated autonomous primal-dual algorithm
  10. Numerical experiments

Key Result

lemma thmcounterlemma

attouchbot2025 Suppose that there exist $C_0>0$ and $b>a \geq 0$ such that where $\tau$ is defined in terms of $\mu$ as in System (dyn), namely, $\tau(t)=\frac{1}{q^q}\left( t_0+\int_{t_0}^t [\mu(s)]^{\frac{1}{q}} ds \right)^q$. Then, there exists $C_1>0$ such that

Theorems & Definitions (12)

  • lemma thmcounterlemma
  • theorem 1
  • proof
  • remark thmcounterremark
  • corollary thmcountercorollary
  • corollary thmcountercorollary
  • remark thmcounterremark
  • corollary thmcountercorollary
  • remark thmcounterremark
  • lemma thmcounterlemma
  • ...and 2 more