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Unified Approach of Interior-Point Algorithms for $$P_*(\kappa )$$ P ∗ ( κ ) -LCPs Using a New Class of Algebraically Equivalent Transformations

Author

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  • Tibor Illés

    (Corvinus University of Budapest)

  • Petra Renáta Rigó

    (Corvinus University of Budapest)

  • Roland Török

    (Corvinus University of Budapest)

Abstract

We propose new short-step interior-point algorithms (IPAs) for solving $$P_*(\kappa )$$ P ∗ ( κ ) -linear complementarity problems (LCPs). In order to define the search directions, we use the algebraic equivalent transformation (AET) technique of the system describing the central path. A novelty of the paper is that we introduce a whole, new class of AET functions for which a unified complexity analysis of the IPAs is presented. This class of functions differs from the ones used in the literature for determining search directions, like the class of concave functions determined by Haddou, Migot and Omer, self-regular functions, eligible kernel and self-concordant functions. We prove that the IPAs using any member $$\varphi $$ φ of the new class of AET functions have polynomial iteration complexity in the size of the problem, in starting point’s duality gap, in the accuracy parameter and in the parameter $$\kappa $$ κ .

Suggested Citation

  • Tibor Illés & Petra Renáta Rigó & Roland Török, 2024. "Unified Approach of Interior-Point Algorithms for $$P_*(\kappa )$$ P ∗ ( κ ) -LCPs Using a New Class of Algebraically Equivalent Transformations," Journal of Optimization Theory and Applications, Springer, vol. 202(1), pages 27-49, July.
    • Handle: RePEc:spr:joptap:v:202:y:2024:i:1:d:10.1007_s10957-023-02232-1
    DOI: 10.1007/s10957-023-02232-1
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    References listed on IDEAS

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    1. H. Mansouri & M. Pirhaji, 2013. "A Polynomial Interior-Point Algorithm for Monotone Linear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 157(2), pages 451-461, May.
    2. 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.
    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.
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