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A new long-step interior point algorithm for linear programming based on the algebraic equivalent transformation

Author

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  • Marianna E.-Nagy

    (Budapest University of Technology and Economics
    Corvinus University of Budapest)

  • Anita Varga

    (Budapest University of Technology and Economics)

Abstract

In this paper, we investigate a new primal-dual long-step interior point algorithm for linear optimization. Based on the step size, interior point algorithms can be divided into two main groups, short-step, and long-step methods. In practice, long-step variants perform better, but usually, a better theoretical complexity can be achieved for the short-step methods. One of the exceptions is the large-update algorithm of Ai and Zhang. The new wide neighborhood and the main characteristics of the presented algorithm are based on their approach. In addition, we use the algebraic equivalent transformation technique of Darvay to determine new modified search directions for our method. We show that the new long-step algorithm is convergent and has the best known iteration complexity of short-step variants. We present our numerical results and compare the performance of our algorithm with two previously introduced Ai-Zhang type interior point algorithms on a set of linear programming test problems from the Netlib library.

Suggested Citation

  • Marianna E.-Nagy & Anita Varga, 2023. "A new long-step interior point algorithm for linear programming based on the algebraic equivalent transformation," 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. 31(3), pages 691-711, September.
    • Handle: RePEc:spr:cejnor:v:31:y:2023:i:3:d:10.1007_s10100-022-00812-6
    DOI: 10.1007/s10100-022-00812-6
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    References listed on IDEAS

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    1. Ilan Adler & Narendra Karmarkar & Mauricio G. C. Resende & Geraldo Veiga, 1989. "Data Structures and Programming Techniques for the Implementation of Karmarkar's Algorithm," INFORMS Journal on Computing, INFORMS, vol. 1(2), pages 84-106, May.
    2. 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.
    3. Nikolaos Ploskas & Nikolaos Samaras, 2017. "Correction to: Linear Programming Using MATLAB®," Springer Optimization and Its Applications, in: Linear Programming Using MATLAB®, pages E1-E3, Springer.
    4. 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.
    5. Nikolaos Ploskas & Nikolaos Samaras, 2017. "Linear Programming Using MATLAB®," Springer Optimization and Its Applications, Springer, number 978-3-319-65919-0, 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. Janez Povh & Lidija Zadnik Stirn & Janez Žerovnik, 2023. "60 years of OR in Slovenia: development from a first conference to a vibrant community," 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. 31(3), pages 681-690, September.

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