Non-ergodic convergence rate of an inertial accelerated primal-dual algorithm for saddle point problems
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| Format: | Preprint |
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2023
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| _version_ | 1866917640554938368 |
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| author | He, X. Huang, N. J. Fang, Y. P. |
| author_facet | He, X. Huang, N. J. Fang, Y. P. |
| contents | In this paper, we design an inertial accelerated primal-dual algorithm to address the convex-concave saddle point problem, which is formulated as $\min_{x}\max_{y} f(x) + \langle Kx, y \rangle - g(y)$. Remarkably, both functions $f$ and $g$ exhibit a composite structure, combining ``nonsmooth'' + ``smooth'' components. Under the assumption of partially strong convexity in the sense that $f$ is convex and $g$ is strongly convex, we introduce a novel inertial accelerated primal-dual algorithm based on Nesterov's extrapolation. This algorithm can be reduced to two classical accelerated forward-backward methods for unconstrained optimization problem. We show that the proposed algorithm achieves a non-ergodic $\mathcal{O}(1/k^2)$ convergence rate, where $k$ represents the number of iterations. Several numerical experiments validate the efficiency of our proposed algorithm. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_11274 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Non-ergodic convergence rate of an inertial accelerated primal-dual algorithm for saddle point problems He, X. Huang, N. J. Fang, Y. P. Optimization and Control In this paper, we design an inertial accelerated primal-dual algorithm to address the convex-concave saddle point problem, which is formulated as $\min_{x}\max_{y} f(x) + \langle Kx, y \rangle - g(y)$. Remarkably, both functions $f$ and $g$ exhibit a composite structure, combining ``nonsmooth'' + ``smooth'' components. Under the assumption of partially strong convexity in the sense that $f$ is convex and $g$ is strongly convex, we introduce a novel inertial accelerated primal-dual algorithm based on Nesterov's extrapolation. This algorithm can be reduced to two classical accelerated forward-backward methods for unconstrained optimization problem. We show that the proposed algorithm achieves a non-ergodic $\mathcal{O}(1/k^2)$ convergence rate, where $k$ represents the number of iterations. Several numerical experiments validate the efficiency of our proposed algorithm. |
| title | Non-ergodic convergence rate of an inertial accelerated primal-dual algorithm for saddle point problems |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2311.11274 |