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An alternative extrapolation scheme of PDHGM for saddle point problem with nonlinear function

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  • Ying Gao

    (University of Electronic Science and Technology of China)

  • Wenxing Zhang

    (University of Electronic Science and Technology of China)

Abstract

Primal-dual hybrid gradient (PDHG) method is a canonical and popular prototype for solving saddle point problem (SPP). However, the nonlinear coupling term in SPP excludes the application of PDHG on far-reaching real-world problems. In this paper, following the seminal work by Valkonen (Inverse Problems 30, 2014), we devise a variant iterative scheme for solving SPP with nonlinear function by exerting an alternative extrapolation procedure. The novel iterative scheme falls exactly into the proximal point algorithmic framework without any residuals, which indicates that the associated inclusion problem is “nearer” to the KKT mapping induced by SPP. Under the metrically regular assumption on KKT mapping, we simplify the local convergence of the proposed method on contractive perspective. Numerical simulations on a PDE-constrained nonlinear inverse problem demonstrate the compelling performance of the proposed method.

Suggested Citation

  • Ying Gao & Wenxing Zhang, 2023. "An alternative extrapolation scheme of PDHGM for saddle point problem with nonlinear function," Computational Optimization and Applications, Springer, vol. 85(1), pages 263-291, May.
    • Handle: RePEc:spr:coopap:v:85:y:2023:i:1:d:10.1007_s10589-023-00453-8
    DOI: 10.1007/s10589-023-00453-8
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    References listed on IDEAS

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    1. Guoyong Gu & Bingsheng He & Xiaoming Yuan, 2014. "Customized proximal point algorithms for linearly constrained convex minimization and saddle-point problems: a unified approach," Computational Optimization and Applications, Springer, vol. 59(1), pages 135-161, October.
    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.
    3. Bingsheng He & Xiaoming Yuan & Wenxing Zhang, 2013. "A customized proximal point algorithm for convex minimization with linear constraints," Computational Optimization and Applications, Springer, vol. 56(3), pages 559-572, December.
    4. Xingju Cai & Deren Han & Lingling Xu, 2013. "An improved first-order primal-dual algorithm with a new correction step," Journal of Global Optimization, Springer, vol. 57(4), pages 1419-1428, December.
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