Continuous and discrete-time accelerated methods for an inequality constrained convex optimization problem
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866917845034598400 |
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| author | Liu, Juan Huang, Nan-Jing Long, Xian-Jun Li, Xue-song |
| author_facet | Liu, Juan Huang, Nan-Jing Long, Xian-Jun Li, Xue-song |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_14828 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Continuous and discrete-time accelerated methods for an inequality constrained convex optimization problem Liu, Juan Huang, Nan-Jing Long, Xian-Jun Li, Xue-song Optimization and Control 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. |
| title | Continuous and discrete-time accelerated methods for an inequality constrained convex optimization problem |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2411.14828 |