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arXiv · 2411.14828

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

Juan Liu·Nan-Jing Huang·Xian-Jun Long·Xue-song Li
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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.

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Juan Liu, Nan-Jing Huang, Xian-Jun Long, Xue-song Li. 2024-11-22. Continuous and discrete-time accelerated methods for an inequality constrained convex optimization problem. https://arxiv.org/abs/2411.14828

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