Journal of the Operations Research Society of China >

2023 , Vol. 11 >Issue 2: 383 - 408

DOI: https://doi.org/10.1007/s40305-022-00412-w

Special Issue: Machine Learning and Optimization Algorithm

Approximate Customized Proximal Point Algorithms for Separable Convex Optimization

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  • 1 School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, Jiangsu, China;
    2 Jiangsu Key Laboratory for NSLSCS, Nanjing Normal University, Nanjing 210023, Jiangsu, China

Received date: 2021-07-01

  Revised date: 2022-02-21

  Online published: 2023-05-24

Supported by

the National Natural Science Foundation of China (Nos. 11971238 and 11871279)

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Abstract

Proximal point algorithm (PPA) is a useful algorithm framework and has good convergence properties. Themain difficulty is that the subproblems usually only have iterative solutions. In this paper, we propose an inexact customized PPA framework for twoblock separable convex optimization problem with linear constraint. We design two types of inexact error criteria for the subproblems. The first one is absolutely summable error criterion, under which both subproblems can be solved inexactly. When one of the two subproblems is easily solved, we propose another novel error criterion which is easier to implement, namely relative error criterion. The relative error criterion only involves one parameter, which is more implementable. We establish the global convergence and sub-linear convergence rate in ergodic sense for the proposed algorithms. The numerical experiments on LASSO regression problems and total variation-based image denoising problem illustrate that our new algorithms outperform the corresponding exact algorithms.

Key words： Inexact criteria; Proximal point algorithm; Alternating direction method of multipliers; Separable convex programming

Cite this article

Hong-Mei Chen, Xing-Ju Cai, Ling-Ling Xu . Approximate Customized Proximal Point Algorithms for Separable Convex Optimization[J]. Journal of the Operations Research Society of China, 2023 , 11(2) : 383 -408 . DOI: 10.1007/s40305-022-00412-w

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