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Sufficient Matrices: Properties, Generating and Testing

Author

Listed:
  • Marianna E.-Nagy

    (Corvinus University of Budapest)

  • Tibor Illés

    (Corvinus University of Budapest)

  • Janez Povh

    (University of Ljubljana)

  • Anita Varga

    (Budapest University of Technology and Economics)

  • Janez Žerovnik

    (University of Ljubljana)

Abstract

This paper investigates various aspects of sufficient matrices, one of the most relevant matrix classes introduced in connection with linear complementarity problems. We summarize the most important theoretical results and properties related to sufficient matrices. Based on these, we propose different construction rules that can be used to generate new matrices that belong to this class. A nonnegative number can be assigned to each sufficient matrix, which is called its handicap and works as a measure of sufficiency. The handicap plays a crucial role in proving convergence and complexity results for interior point algorithms for linear complementarity problems. For a particular sufficient matrix, called Csizmadia’s matrix, we give the exact value of the handicap, which is exponential in the size of the matrix. Another important topic that we address is deciding whether a matrix is sufficient. Tseng proved in 2000 that this decision problem is co-NP hard. We investigate three different algorithms for determining the sufficiency of a given matrix: Väliaho’s algorithm, a linear programming-based algorithm, and an algorithm that facilitates nonlinear programming reformulations of the definition of sufficiency. We tested the efficiency of these methods on our recently launched benchmark data set that consists of four different sets of matrices. In this paper, we give the description and most important properties of the benchmark set, which can be used in the future to compare the performance of different interior point algorithms for linear complementarity problems.

Suggested Citation

  • 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.
  • Handle: RePEc:spr:joptap:v:202:y:2024:i:1:d:10.1007_s10957-023-02280-7
    DOI: 10.1007/s10957-023-02280-7
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    References listed on IDEAS

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    1. 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.
    2. T. Illés & M. Nagy & T. Terlaky, 2009. "EP Theorem for Dual Linear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 140(2), pages 233-238, February.
    3. Illes, Tibor & Nagy, Marianna, 2007. "A Mizuno-Todd-Ye type predictor-corrector algorithm for sufficient linear complementarity problems," European Journal of Operational Research, Elsevier, vol. 181(3), pages 1097-1111, September.
    4. Janez Povh & Janez Žerovnik, 2021. "On sufficient properties of sufficient matrices," 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. 29(3), pages 809-822, September.
    5. 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.
    6. 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.
    7. Tibor Illés & Marianna Nagy & Tamás Terlaky, 2010. "A polynomial path-following interior point algorithm for general linear complementarity problems," Journal of Global Optimization, Springer, vol. 47(3), pages 329-342, July.
    8. Illes, Tibor & Terlaky, Tamas, 2002. "Pivot versus interior point methods: Pros and cons," European Journal of Operational Research, Elsevier, vol. 140(2), pages 170-190, July.
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