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On solving a larger subclass of linear complementarity problems by Lemke’s method

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  • Sajal Ghosh

    (Indian Statistical Institute)

  • Gambheer Singh

    (Indian Statistical Institute)

  • Deepayan Sarkar

    (Indian Statistical Institute)

  • S. K. Neogy

    (Indian Statistical Institute)

Abstract

It is well known that the success of Lemke’s algorithm for solving a linear complementarity problem LCP(q, M) depends on the matrix class M. Many researchers investigated a large class of matrices for which Lemke’s algorithm computes a solution of the LCP(q, M). In this paper, we follow a different approach for the class of LCP, which is not solvable by Lemke’s algorithm. First, we construct an artificial LCP $$(\bar{q}_{1},\mathcal {M}_{1})$$ ( q ¯ 1 , M 1 ) from LCP(q, M) by adding some artificial variables and extra constraints, and show that the matrix $$\mathcal {M}_{1}$$ M 1 belongs to the class of semimonotone matrices. However, LCP $$(\bar{q}_{1},\mathcal {M}_{1})$$ ( q ¯ 1 , M 1 ) is not always solvable by Lemke’s algorithm. Then, we construct another artificial LCP $$(\bar{q}_{2},\mathcal {M}_{2})$$ ( q ¯ 2 , M 2 ) from LCP $$(\bar{q}_{1},\mathcal {M}_{1})$$ ( q ¯ 1 , M 1 ) by adding some more artificial variables and extra constraints that satisfy Eaves condition. We show that the resulting artificial LCP $$(\bar{q}_{2},\mathcal {M}_{2})$$ ( q ¯ 2 , M 2 ) is solvable by Lemke’s algorithm. Given an LCP(q, M), its solution can be obtained from the solution of the constructed artificial LCP( $$\bar{q}_{2},\mathcal {M}_{2}$$ q ¯ 2 , M 2 ) with Eaves conditions. This approach leads to an innovative scheme for solving a large class of LCPs which are not solvable by Lemke’s algorithm. Further, we also provide convergence results. The results obtained here can be used for broader applications of Lemke’s algorithm.

Suggested Citation

  • Sajal Ghosh & Gambheer Singh & Deepayan Sarkar & S. K. Neogy, 2025. "On solving a larger subclass of linear complementarity problems by Lemke’s method," Indian Journal of Pure and Applied Mathematics, Springer, vol. 56(3), pages 1014-1025, September.
    • Handle: RePEc:spr:indpam:v:56:y:2025:i:3:d:10.1007_s13226-025-00817-2
    DOI: 10.1007/s13226-025-00817-2
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    References listed on IDEAS

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    1. Michael J. Todd, 1976. "Orientation in Complementary Pivot Algorithms," Mathematics of Operations Research, INFORMS, vol. 1(1), pages 54-66, February.
    2. C. E. Lemke, 1965. "Bimatrix Equilibrium Points and Mathematical Programming," Management Science, INFORMS, vol. 11(7), pages 681-689, May.
    3. B. Curtis Eaves, 1971. "The Linear Complementarity Problem," Management Science, INFORMS, vol. 17(9), pages 612-634, May.
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