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A New Ai–Zhang Type Interior Point Algorithm for Sufficient Linear Complementarity Problems

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  • Marianna E.-Nagy

    (Corvinus University of Budapest)

  • Anita Varga

    (Budapest University of Technology and Economics)

Abstract

In this paper, we propose a new long-step interior point method for solving sufficient linear complementarity problems. The new algorithm combines two important approaches from the literature: the main ideas of the long-step interior point algorithm introduced by Ai and Zhang and the algebraic equivalent transformation technique proposed by Darvay. Similar to the method of Ai and Zhang, our algorithm also works in a wide neighborhood of the central path and has the best known iteration complexity of short-step variants. However, due to the properties of the applied transforming function in Darvay’s technique, the wide neighborhood definition in the analysis depends on the value of the handicap. We implemented not only the theoretical algorithm but a greedy variant of the new method (working in a neighborhood independent of the handicap) in MATLAB and tested its efficiency on both sufficient and non-sufficient problem instances. In addition to presenting our numerical results, we also make some interesting observations regarding the analysis of Ai–Zhang type methods.

Suggested Citation

  • Marianna E.-Nagy & Anita Varga, 2024. "A New Ai–Zhang Type Interior Point Algorithm for Sufficient Linear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 202(1), pages 76-107, July.
    • Handle: RePEc:spr:joptap:v:202:y:2024:i:1:d:10.1007_s10957-022-02121-z
    DOI: 10.1007/s10957-022-02121-z
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    References listed on IDEAS

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    1. Y. Q. Bai & G. Lesaja & C. Roos & G. Q. Wang & M. El Ghami, 2008. "A Class of Large-Update and Small-Update Primal-Dual Interior-Point Algorithms for Linear Optimization," Journal of Optimization Theory and Applications, Springer, vol. 138(3), pages 341-359, September.
    2. Filiz Gurtuna & Cosmin Petra & Florian Potra & Olena Shevchenko & Adrian Vancea, 2011. "Corrector-predictor methods for sufficient linear complementarity problems," Computational Optimization and Applications, Springer, vol. 48(3), pages 453-485, April.
    3. Soodabeh Asadi & Hossein Mansouri & Zsolt Darvay & Maryam Zangiabadi & Nezam Mahdavi-Amiri, 2019. "Large-Neighborhood Infeasible Predictor–Corrector Algorithm for Horizontal Linear Complementarity Problems over Cartesian Product of Symmetric Cones," Journal of Optimization Theory and Applications, Springer, vol. 180(3), pages 811-829, March.
    4. Illés, Tibor & Rigó, Petra Renáta & Török, Roland, 2021. "Predictor-corrector interior-point algorithm based on a new search direction working in a wide neighbourhood of the central path," Corvinus Economics Working Papers (CEWP) 2021/02, Corvinus University of Budapest.
    5. Zsolt Darvay & Petra Renáta Takács, 2018. "Large-step interior-point algorithm for linear optimization based on a new wide neighbourhood," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 26(3), pages 551-563, September.
    6. Zs. Darvay & T. Illés & B. Kheirfam & P. R. Rigó, 2020. "A corrector–predictor interior-point method with new search direction for linear optimization," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 28(3), pages 1123-1140, September.
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    Cited by:

    1. Marianna E.-Nagy & Tibor Illés & Janez Povh & Anita Varga & Janez Žerovnik, 2024. "Sufficient Matrices: Properties, Generating and Testing," Journal of Optimization Theory and Applications, Springer, vol. 202(1), pages 204-236, July.

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    More about this item

    Keywords

    Mathematical programming; Linear complementarity problems; Interior point algorithms; Algebraic equivalent transformation technique;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

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