Perturbed Proximal Gradient ADMM for Nonconvex Composite Optimization
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911351357571072 |
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| author | Zhou, Yuan Shi, Xinli Guo, Luyao Cao, Jinde Abdel-Aty, Mahmoud |
| author_facet | Zhou, Yuan Shi, Xinli Guo, Luyao Cao, Jinde Abdel-Aty, Mahmoud |
| contents | This paper proposes a Perturbed Proximal Gradient ADMM (PPG-ADMM) framework for solving general nonconvex composite optimization problems, where the objective function consists of a smooth nonconvex term and a nonsmooth weakly convex term for both primal variables.
Unlike existing ADMM-based methods which necessitate the function associated with the last updated primal variable to be smooth, the proposed PPG-ADMM removes this restriction by introducing a perturbation mechanism, which also helps reduce oscillations in the primal-dual updates, thereby improving convergence stability.
By employing a linearization technique for the smooth term and the proximal operator for the nonsmooth and weakly convex term, the subproblems have closed-form solutions, significantly reducing computational complexity. The convergence is established through a technically constructed Lyapunov function, which guarantees sufficient descent and has a well-defined lower bound.
With properly chosen parameters, PPG-ADMM converges to an $ε$-approximate stationary point at a sublinear convergence rate of $\mathcal{O}(1/\sqrt{K})$.
Furthermore, by appropriately tuning the perturbation parameter $β$, it achieves an $ε$-stationary point, providing stronger optimality guarantees. We further apply PPG-ADMM to two practical distributed nonconvex composite optimization problems, i.e., the distributed partial consensus problem and the resource allocation problem. The algorithm operates in a fully decentralized manner without a central coordinating node. Finally, numerical experiments validate the effectiveness of PPG-ADMM, demonstrating its improved convergence performance. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_12759 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Perturbed Proximal Gradient ADMM for Nonconvex Composite Optimization Zhou, Yuan Shi, Xinli Guo, Luyao Cao, Jinde Abdel-Aty, Mahmoud Optimization and Control This paper proposes a Perturbed Proximal Gradient ADMM (PPG-ADMM) framework for solving general nonconvex composite optimization problems, where the objective function consists of a smooth nonconvex term and a nonsmooth weakly convex term for both primal variables. Unlike existing ADMM-based methods which necessitate the function associated with the last updated primal variable to be smooth, the proposed PPG-ADMM removes this restriction by introducing a perturbation mechanism, which also helps reduce oscillations in the primal-dual updates, thereby improving convergence stability. By employing a linearization technique for the smooth term and the proximal operator for the nonsmooth and weakly convex term, the subproblems have closed-form solutions, significantly reducing computational complexity. The convergence is established through a technically constructed Lyapunov function, which guarantees sufficient descent and has a well-defined lower bound. With properly chosen parameters, PPG-ADMM converges to an $ε$-approximate stationary point at a sublinear convergence rate of $\mathcal{O}(1/\sqrt{K})$. Furthermore, by appropriately tuning the perturbation parameter $β$, it achieves an $ε$-stationary point, providing stronger optimality guarantees. We further apply PPG-ADMM to two practical distributed nonconvex composite optimization problems, i.e., the distributed partial consensus problem and the resource allocation problem. The algorithm operates in a fully decentralized manner without a central coordinating node. Finally, numerical experiments validate the effectiveness of PPG-ADMM, demonstrating its improved convergence performance. |
| title | Perturbed Proximal Gradient ADMM for Nonconvex Composite Optimization |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2504.12759 |