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<p>The proximal point algorithm (PPA) stands as a fundamental approach for solving monotone inclusion problems. Notably, several key convex optimization algorithms have been proven to be specific instances of PPA. Given the importance of the PPA, there has been growing interest in developing its accelerated variants. However, some existing accelerated PPAs exhibit oscillatory behavior, which can impede their numerical convergence rate. In this paper, we first introduce an ODE system and demonstrate its <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( o(1/t^2) \)</EquationSource> </InlineEquation> convergence rate and weak convergence property. Next, we apply the Symplectic Euler Method to discretize the ODE and obtain a new accelerated PPA, which we call the symplectic proximal point algorithm (SPPA). The reason for using the Symplectic Euler Method is its ability to preserve the geometric structure of the ODEs. Theoretically, we demonstrate that the SPPA achieves an <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\( o(1/k^2) \)</EquationSource> </InlineEquation> convergence rate and that the sequences generated by our method converge weakly to the solution set. Practically, our numerical experiments illustrate that the SPPA significantly reduces oscillatory behavior, leading to improved long-time behavior and faster numerical convergence rate.</p>

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Symplectic discretization approach for developing new proximal point algorithm

  • 期刊论文

  • 发表日期：

    2025年11月10日

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  • Ya-xiang Yuan,
  • Yi Zhang

摘要

The proximal point algorithm (PPA) stands as a fundamental approach for solving monotone inclusion problems. Notably, several key convex optimization algorithms have been proven to be specific instances of PPA. Given the importance of the PPA, there has been growing interest in developing its accelerated variants. However, some existing accelerated PPAs exhibit oscillatory behavior, which can impede their numerical convergence rate. In this paper, we first introduce an ODE system and demonstrate its \( o(1/t^2) \) convergence rate and weak convergence property. Next, we apply the Symplectic Euler Method to discretize the ODE and obtain a new accelerated PPA, which we call the symplectic proximal point algorithm (SPPA). The reason for using the Symplectic Euler Method is its ability to preserve the geometric structure of the ODEs. Theoretically, we demonstrate that the SPPA achieves an \( o(1/k^2) \) convergence rate and that the sequences generated by our method converge weakly to the solution set. Practically, our numerical experiments illustrate that the SPPA significantly reduces oscillatory behavior, leading to improved long-time behavior and faster numerical convergence rate.