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Anderson accelerating the preconditioned modulus approach for linear complementarity problems on second-order cones

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Abstract

Second-order cone linear complementarity problems (SOCLCPs) have wide applications in real world, and the latest modulus method is proved to be an efficient solver. Here, inspired by the state-of-the-art modulus method and Anderson acceleration (AA), we construct the Anderson accelerating preconditioned modulus (AA+PMS) approach. Theoretically, in the first stage, we utilize the Fr\(\acute { \mathrm {e}}\)chet-differentiability of the absolute value function in Jordan algebra to explore its new properties. On this basis, we establish the convergence theory for the PMS approach different from the previous analysis, and further discuss the selection strategy of parameters involved. In the second stage, we demonstrate the strong semi-smoothness of the absolute value function in Jordan algebra and, thus, establish the local convergence theory for the AA+PMS approach. Finally, we conduct rich numerical experiments with application to some well-structured examples, the second-order cone programming, the Signorini problem of the Laplacian and the three-dimensional frictional contact problem to verify the robustness and effectiveness of the AA+PMS approach.

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Acknowledgements

We are very grateful to anonymous reviewers for their valuable comments that make our work better.

Funding

This paper is supported by National Science Foundation of China (Nos. 42004085, 41725017, 62173235 and 61602309), China Postdoctoral Science Foundation (No. 2019M663040), Guangdong Basic and Applied Basic Research Foundation (Nos. 2019A1515110184), the National Key R & D Program of the Ministry of Science and Technology of China (No. 2020YFA0713400 and 2020YFA0713401).

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  1. Guangdong Key Laboratory of Intelligent Information Processing and Shenzhen Key Laboratory of Media Security, College of Electronics and Information Engineering, Shenzhen University, Shenzhen, 518060, China

    Zhizhi Li & Le Ou-Yang

  2. Key Laboratory of Computational Geodynamics, University of Chinese Academy of Sciences, Beijing, 100049, China

    Zhizhi Li, Huai Zhang & Yimin Jin

  3. Laboratory for Marine Mineral Resources, Qingdao National Laboratory for Marine Science and Technology, Qingdao, 266237, China

    Huai Zhang

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  1. Zhizhi Li
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Li, Z., Zhang, H., Jin, Y. et al. Anderson accelerating the preconditioned modulus approach for linear complementarity problems on second-order cones. Numer Algor 91, 803–839 (2022). https://doi.org/10.1007/s11075-022-01283-1

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