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An Efficient Interior Point Method for Linear Optimization Using Modified Newton Method

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Algorithms and Discrete Applied Mathematics (CALDAM 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14508))

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Abstract

Interior point methods are one of the popular iterative approaches for solving optimization problems. Search direction plays a vital role in the performance of the interior point methods. This paper uses a modification to the Newton method and proposes a new way to find the search direction. We introduce a two-step interior point algorithm for solving linear optimization problems based on the new search direction. We present theoretical results for the convergence of the algorithm. Finally, we evaluate the algorithm on some test problems from the Netlib collection and show that the proposed algorithm reduces the number of iterations and CPU time by \(30.97\%\) and \(20.46\%\), respectively.

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Notes

  1. 1.

    \(\Vert x\Vert _1=\sum _{i=1}^n|x_i|\).

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Authors and Affiliations

  1. Department of Math and Computer Science, University of Lethbridge, Lethbridge, Canada

    Sajad Fathi Hafshejani, Daya Gaur & Robert Benkoczi

Authors
  1. Sajad Fathi Hafshejani
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  2. Daya Gaur
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  3. Robert Benkoczi
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Corresponding author

Correspondence to Sajad Fathi Hafshejani .

Editor information

Editors and Affiliations

  1. Indian Institute of Technology Hyderabad, Sangareddy, India

    Subrahmanyam Kalyanasundaram

  2. Carleton University, Ottawa, ON, Canada

    Anil Maheshwari

Appendix

Appendix

The Appendix section contains two tables. Table 2 specifies information related to each test problem, including the name, the number of non-zero elements, and the number of rows and columns of matrix A. Table 3 provides information on the number of iterations and the CPU time for performing Algorithm 1 and the classical algorithm proposed in [12]. Additionally, the averages of CPU time and the number of iterations from Table 3 are presented in Table 1.

Table 2. LP instances from NETLIB
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Table 3. Number of iterations and CPU times for classical algorithm and Algorithm 1.
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© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG

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Hafshejani, S.F., Gaur, D., Benkoczi, R. (2024). An Efficient Interior Point Method for Linear Optimization Using Modified Newton Method. In: Kalyanasundaram, S., Maheshwari, A. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2024. Lecture Notes in Computer Science, vol 14508. Springer, Cham. https://doi.org/10.1007/978-3-031-52213-0_10

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  • DOI: https://doi.org/10.1007/978-3-031-52213-0_10

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-52212-3

  • Online ISBN: 978-3-031-52213-0

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