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The Developments of Proximal Point Algorithms

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Abstract

The problem of finding a zero point of a maximal monotone operator plays a central role in modeling many application problems arising from various fields, and the proximal point algorithm (PPA) is among the fundamental algorithms for solving the zero-finding problem. PPA not only provides a very general framework of analyzing convergence and rate of convergence of many algorithms, but also can be very efficient in solving some structured problems. In this paper, we give a survey on the developments of PPA and its variants, including the recent results with linear proximal term, with the nonlinear proximal term, as well as the inexact forms with various approximate criteria.

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Notes

  1. A pair of vectors \(x_{*}\in {\mathbb {R}}^n\) and \(\lambda _{*}\in {\mathbb {R}}^m\) is called a saddle point of \({\mathcal {L}}\) if

    $$\begin{aligned} {\mathcal {L}}(x_{*},\lambda )\leqslant {\mathcal {L}}(x_{*},\lambda _{*})\leqslant {\mathcal {L}}(x,\lambda _{*}),\qquad \forall x\in {\mathbb {R}}^n,\;\forall \lambda \in {\mathbb {R}}^m. \end{aligned}$$

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Authors and Affiliations

  1. Jiangsu Key Lab for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, Jiangsu, China

    Xing-Ju Cai

  2. School of Mathematics and Information, China West Normal University, Nanchong, 637002, Sichuan, China

    Ke Guo

  3. Department of Information and Computing Science, School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, Jiangsu, China

    Fan Jiang

  4. Department of Mathematics, Nanjing University of Science and Technology, Nanjing, 210094, Jiangsu, China

    Kai Wang

  5. School of Management Science and Engineering, Nanjing University of Information Science and Technology, Nanjing, 210044, Jiangsu, China

    Zhong-Ming Wu

  6. LMIB, School of Mathematical Sciences, Beihang University, Beijing, 100191, China

    De-Ren Han

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Xing-Ju Cai and Fan Jiang were supported by the National Natural Science Foundation of China (Nos. 11871279 and 11571178). Ke Guo was supported by the National Natural Science Foundation of China (Nos. 11801455, 11871059 and 11971238), China Postdoctoral Science Foundation (Nos. 2019M663459 and 2020T130081), the Applied Basic Project of Sichuan Province (No. 2020YJ0111), the Fundamental Research Funds of China West Normal University (No. 18B031) and the Open Project of Key Laboratory (No. CSSXKFKTM202004), School of Mathematical Sciences, Chongqing Normal University. Kai Wang was supported by the National Natural Science Foundation of China (No. 11901294) and Natural Science Foundation of Jiangsu Province (No. BK20190429). Zhong-Ming Wu was supported by the National Natural Science Foundation of China (No. 12001286) and the Startup Foundation for Introducing Talent of NUIST (No. 2020r003). De-Ren Han was supported by the National Natural Science Foundation of China (Nos. 12131004 and 12126603).

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Cai, XJ., Guo, K., Jiang, F. et al. The Developments of Proximal Point Algorithms. J. Oper. Res. Soc. China 10, 197–239 (2022). https://doi.org/10.1007/s40305-021-00352-x

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