Advertisement

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

  • Xiang Wang
  • Xing Li
  • Lei-Hong Zhang
  • Ren-Cang LiEmail author
Article
First Online:
  • 31 Downloads

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.

Keywords

Second-order cone Linear complementarity problem SOCLCP Globally uniquely solvable property GUS 

Mathematics Subject Classification

90C33 65K05 65F99 65F15 65F30 65P99 
This is a preview of subscription content, log in to check access.

Notes

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.

References

  1. 1.
    Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Progr. 95, 3–51 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Anderson, E., Bai, Z., Bischof, C., Demmel, J.W., Dongarra, J., Croz, J.D., Greenbaum, A., Hammarling, S., McKenney, A., Ostrouchov, S., Sorensen, D.: LAPACK Users’ Guide, 3rd edn. SIAM, Philadelphia (1999)CrossRefzbMATHGoogle Scholar
  3. 3.
    Auslender, A.: Variational inequalities over the cone of semidefinite positive symmetric matrices and over the Lorentz cone. Opt. Methods Soft. 18, 359–376 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, Cambridge (1982)zbMATHGoogle Scholar
  5. 5.
    Brás, C.P., Fischerb, A., Júdice, J.J., Schönefeldb, K., Seifert, S.: A block active set algorithm with spectral choice line search for the symmetric eigenvalue complementarity problem. Appl. Math. Comput. 294, 36–48 (2017)MathSciNetGoogle Scholar
  6. 6.
    Bunch, J.R., Nielsen, C.P., Sorensen, D.C.: Rank-one modification of the symmetric eigenproblem. Numer. Math. 31, 31–48 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chen, J.S., Tseng, P.: An unconstrained smooth minimization reformulation of the second-order cone complementarity problem. Math. Progr. 104, 293–327 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chen, X.D., Sun, D.F., Sun, J.: Complementarity functions and numerical experiments on some smoothing Newton methods for second-order-cone complementarity problems. Comput. Optim. Appl. 25, 39–56 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cuppen, J.J.M.: A divide and conquer method for the symmetric eigenproblem. Numer. Math. 36, 177–195 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fernandes, L.M., Fukushima, M., Júdice, J.J., Sherali, H.D.: The second-order cone eigenvalue complementarity problem. Opt. Methods Soft. 31, 24–52 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fukushima, M., Luo, Z.Q., Tseng, P.: Smoothing functions for second-order-cone complementarity problems. SIAM J. Optim. 12, 436–460 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Goldfarb, D., Scheinberg, K.: Product-form cholesky factorization in interior point methods for second-order cone programming. Math. Progr. 103, 153–179 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gowda, M.S., Sznajder, R.: Automorphism invariance of P- and GUS-properties of linear transformations on Euclidean Jordan algebras. Math. Oper. Res. 31, 109–123 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gowda, M.S., Sznajder, R.: Some global uniqueness and solvability results for linear complementarity problems over symmetric cones. SIAM J. Optim. 18, 461–481 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gowda, M.S., Sznajder, R., Tao, J.: Some P-properties for linear transformations on Euclidean Jordan algebras. Linear Algebra Appl. 393, 203–232 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hayashi, S., Yamashita, N., Fukushima, M.: A combined smoothing and regularization method for monotone second-order cone complementarity problems. SIAM J. Optim. 15, 593–615 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hayashi, S., Yamashita, N., Fukushima, M.: A matrix-splitting method for symmetric affine second-order cone complementarity problems. J. Comput. Appl. Math. 175, 335–353 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hestenes, M.R.: Multiplier and gradient methods. J. Optim. Theory Appl. 4, 303–320 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hiriart-Urruty, J.B., Seeger, A.: A variational approach to copositive matrices. SIAM Rev. 52, 593–629 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kong, L.C., Sun, J., Xiu, N.H.: A regularized smoothing Newton method for symmetric cone complementarity problems. SIAM J. Optim. 19, 1028–1047 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Li, R.-C.: Solving secular equations stably and efficiently. Technical Report UCB//CSD-94-851, Computer Science Division, Department of EECS, University of California at Berkeley (1993)Google Scholar
  22. 22.
    Lobo, M.S., Vandenberghe, L., Boyd, S., Lebret, H.: Application of second-order cone programming. Linear Algebra Appl. 284(2), 193–228 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Loewy, R.: Positive operators on the n-dimensional ice cream cone. J. Math. Anal. Appl. 49(2), 375–392 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Luo, Z.Q., Tseng, P.: On the linear convergence of descent methods for convex essentially smooth minimization. SIAM J. Control Optim. 30(2), 408–425 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Nocedal, J., Wright, S.: Numerical Optimization, 2nd edn. Springer, New York (2006)zbMATHGoogle Scholar
  26. 26.
    Pang, J.S., Sun, D.F., Sun, J.: Semismooth homeomorphisms and strong stability of semidefinite and Lorentz complementarity problems. Math. Oper. Res. 28, 39–63 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Powell, M.J.D.: A method for nonlinear constraints in minimization problems. In: Fletcher, R. (ed.) Optimization, pp. 283–298. Academic Press, New York (1969)Google Scholar
  28. 28.
    Stewart, G.W., Sun, J.G.: Matrix Perturbation Theory. Academic Press, Boston (1990)zbMATHGoogle Scholar
  29. 29.
    Sturm, J.F.: Using SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones. Opt. Methods Soft. 11, 625–653 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Toh, K.C., Todd, M.J., Tütüncü, R.H.: SDPT3-a MATLAB software package for semidefinite programming. Opt. Methods Soft. 11, 545–581 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Tütüncü, R.H., Toh, K.C., Todd, M.J.: Solving semidefinite-quadratic-linear programs using SDPT3. Math. Progr. 95, 189–217 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Yang, W.H., Yuan, X.M.: The GUS-property of second-order cone linear complementarity problems. Math. Progr. 141, 295–317 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Yang, W.H., Zhang, L.-H., Shen, C.: Solution analysis for the pseudomonotone second-order cone linear complementarity problem. Optimization 65(9), 1703–1715 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Yang, W.H., Zhang, L.-H., Shen, C.: On the range of the pseudomonotone second-order cone linear complementarity problem. J. Optim. Theory Appl. 173(2), 504–522 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Zhang, L.-H., Shen, C., Yang, W.H., Júdice, J.J.: A Lanczos method for large-scale extreme Lorentz eigenvalue problems. SIAM J. Matrix Anal. Appl. 39(2), 611–631 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Zhang, L.-H., Yang, W.H.: An efficient algorithm for second-order cone linear complementarity problems. Math. Comput. 83, 1701–1726 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Zhang, L.-H., Yang, W.H.: An efficient matrix splitting method for the second-order cone complementarity problem. SIAM J. Optim. 24(3), 1178–1205 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Zhang, L.-H., Yang, W.H., Shen, C., Li, R.-C.: A Krylov subspace method for large scale second order cone linear complementarity problem. SIAM J. Sci. Comput. 37(4), A2046–A2075 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsNanchang UniversityNanchangChina
  2. 2.School of MathematicsShanghai University of Finance and EconomicsShanghaiChina
  3. 3.School of Mathematical SciencesSoochow UniversitySuzhouChina
  4. 4.Department of MathematicsUniversity of Texas at ArlingtonArlingtonUSA

Personalised recommendations