Continuous and discrete-time accelerated methods for an inequality constrained convex optimization problem

TL;DR

This paper develops accelerated continuous and discrete-time algorithms for inequality constrained convex optimization using Lyapunov functions, Hamiltonian dynamics, and Bregman Lagrangian, achieving exponential convergence and validated by numerical experiments.

Contribution

It introduces a novel continuous-time dynamical system and derived algorithms with optimal convergence rates for inequality constrained convex problems.

Findings

Continuous-time system converges exponentially to the solution.

Derived discrete algorithms achieve optimal convergence rates.

Numerical experiments confirm the effectiveness of the proposed methods.

Abstract

This paper is devoted to the study of acceleration methods for an inequality constrained convex optimization problem by using Lyapunov functions. We first approximate such a problem as an unconstrained optimization problem by employing the logarithmic barrier function. Using the Hamiltonian principle, we propose a continuous-time dynamical system associated with a Bregman Lagrangian for solving the unconstrained optimization problem. Under certain conditions, we demonstrate that this continuous-time dynamical system exponentially converges to the optimal solution of the inequality constrained convex optimization problem. Moreover, we derive several discrete-time algorithms from this continuous-time framework and obtain their optimal convergence rates. Finally, we present numerical experiments to validate the effectiveness of the proposed algorithms.