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Fast convergence of the primal-dual dynamical system and corresponding algorithms for a nonsmooth bilinearly coupled saddle point problem

Author

Listed:
  • Ke-wei Ding

    (Southwest Minzu University)

  • Jörg Fliege

    (University of Southampton)

  • Phan Tu Vuong

    (University of Southampton)

Abstract

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.

Suggested Citation

  • Ke-wei Ding & Jörg Fliege & Phan Tu Vuong, 2025. "Fast convergence of the primal-dual dynamical system and corresponding algorithms for a nonsmooth bilinearly coupled saddle point problem," Computational Optimization and Applications, Springer, vol. 90(1), pages 151-192, January.
    • Handle: RePEc:spr:coopap:v:90:y:2025:i:1:d:10.1007_s10589-024-00626-z
    DOI: 10.1007/s10589-024-00626-z
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    References listed on IDEAS

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    1. Hedy Attouch & Zaki Chbani & Jalal Fadili & Hassan Riahi, 2022. "Fast Convergence of Dynamical ADMM via Time Scaling of Damped Inertial Dynamics," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 704-736, June.
    2. Laurent Condat, 2013. "A Primal–Dual Splitting Method for Convex Optimization Involving Lipschitzian, Proximable and Linear Composite Terms," Journal of Optimization Theory and Applications, Springer, vol. 158(2), pages 460-479, August.
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