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An Efficient Numerical Method for the Symmetric Positive Definite Second-Order Cone Linear Complementarity Problem

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Abstract

An efficient numerical method for solving a symmetric positive definite second-order cone linear complementarity problem (SOCLCP) is proposed. The method is shown to be more efficient than recently developed iterative methods for small-to-medium sized and dense SOCLCP. Therefore it can serve as an excellent core computational engine in solutions of large scale symmetric positive definite SOCLCP solved by subspace projection methods, solutions of general SOCLCP and the quadratic programming over a Cartesian product of multiple second-order cones, in which small-to-medium sized SOCLCPs have to be solved repeatedly, efficiently, and robustly.

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Notes

  1. Equivalently, \((B-C)+(B-C)^{{{\,\mathrm{T}\,}}}\) is symmetric positive definite.

  2. mexeig is available at: www.math.nus.edu.sg/~matsundf/.

  3. LAPACK’s subroutine dsyevd computes all eigenvalues, and optionally, eigenvectors of a real symmetric matrix. It uses the divide-and-conquer algorithm when eigenvectors are desired. Through its MATLAB interface via mexeig, it computes the eigendecomposition of a symmetric matrix much faster than MATLAB’s eig.

  4. http://cvxr.com/cvx.

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Acknowledgements

The authors are grateful to two anonymous referees for their helpful comments and suggestions that improve the presentation. Wang is supported in part by the National Natural Science Foundation of China: NSFC-11461046, NSF of Jiangxi Province: 20161ACB21005 and 20181ACB20001, Zhang is supported in part by the National Natural Science Foundation of China: NSFC-11671246 and NSFC-91730303, and R.-C. Li is supported in part by NSF Grants: CCF-1527104 and DMS-1719620.

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Authors and Affiliations

  1. Department of Mathematics, Nanchang University, 999 Xuefu Road, Nanchang, 330031, China

    Xiang Wang

  2. School of Mathematics, Shanghai University of Finance and Economics, 777 Guoding Road, Shanghai, 200433, China

    Xing Li

  3. School of Mathematical Sciences, Soochow University, Suzhou, 215006, Jiangsu, China

    Lei-Hong Zhang

  4. Department of Mathematics, University of Texas at Arlington, P.O. Box 19408, Arlington, TX, 76019-0408, USA

    Ren-Cang Li

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  1. Xiang Wang
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Wang, X., Li, X., Zhang, LH. et al. An Efficient Numerical Method for the Symmetric Positive Definite Second-Order Cone Linear Complementarity Problem. J Sci Comput 79, 1608–1629 (2019). https://doi.org/10.1007/s10915-019-00907-4

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