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

Sajal Ghosh (), Gambheer Singh (), Deepayan Sarkar () and S. K. Neogy ()
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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

Indian Journal of Pure and Applied Mathematics, 2025, vol. 56, issue 3, 1014-1025

Abstract: 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.

Keywords: Linear complementarity problem; Lemke’s algorithm; Artificial LCP matrix; $$\textbf{L}$$ L -matrix; 90C33 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s13226-025-00817-2

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