Fast convergence of the primal-dual dynamical system and corresponding algorithms for a nonsmooth bilinearly coupled saddle point problem
This paper studies the convergence rate of a second-order dynamical system associated with a nonsmooth bilinearly coupled convex-concave saddle point problem, as well as the convergence rate of its corresponding discretizations. We derive the convergence rate of the primal-dual gap for the second-or...
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| Vydáno v: | Computational optimization and applications Ročník 90; číslo 1; s. 151 - 192 |
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| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
01.01.2025
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| ISSN: | 0926-6003, 1573-2894 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | This paper studies the convergence rate of a second-order dynamical system associated with a nonsmooth bilinearly coupled convex-concave saddle point problem, as well as the convergence rate of its corresponding discretizations. We derive the convergence rate of the primal-dual gap for the second-order dynamical system with asymptotically vanishing damping term. Based on an implicit discretization scheme, we propose a primal-dual algorithm and provide a non-ergodic convergence rate under a general setting for the inertial parameters when one objective function is continuously differentiable and convex and the other is a proper, convex and lower semicontinuous function. For this algorithm we derive a
$$O\left( 1/k^2 \right) $$
O
1
/
k
2
convergence rate under three classical rules proposed by Nesterov, Chambolle-Dossal and Attouch-Cabot without assuming strong convexity, which is compatible with the results of the continuous-time dynamic system. For the case when both objective functions are continuously differentiable and convex, we further present a primal-dual algorithm based on an explicit discretization. We provide a corresponding non-ergodic convergence rate for this algorithm and show that the sequence of iterates generated weakly converges to a primal-dual optimal solution. Finally, we present numerical experiments that indicate the superior numerical performance of both algorithms. |
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| ISSN: | 0926-6003 1573-2894 |
| DOI: | 10.1007/s10589-024-00626-z |