Complementarity Functions and Numerical Experiments on Some Smoothing Newton Methods for Second-Order-Cone Complementarity Problems

  • X.D. Chen
  • D. Sun
  • J. Sun


Two results on the second-order-cone complementarity problem are presented. We show that the squared smoothing function is strongly semismooth. Under monotonicity and strict feasibility we provide a new proof, based on a penalized natural complementarity function, for the solution set of the second-order-cone complementarity problem being bounded. Numerical results of squared smoothing Newton algorithms are reported.

complementarity function soc smoothing Newton method quadratic convergence 


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  1. 1.
    B. Chen, X. Chen, and C. Kanzow, “A penalized Fischer-Burmeister NCP-function,” Math. Prog., vol. 88, pp. 211-216, 2000.Google Scholar
  2. 2.
    F.H. Clarke, Optimization and Nonsmooth Analysis. Wiley: New York, 1983.Google Scholar
  3. 3.
    J.-P. Crouzeix, “Pseudomonotone variational inequality problems: Existence of solutions,” Math. Prog., vol. 78, pp. 305-314, 1997.Google Scholar
  4. 4.
    J. Faraut and A. Koranyi, Analysis on symmetric cones. Clarendon Press: Oxford, 1994.Google Scholar
  5. 5.
    L. Faybusovich, “Linear systems in Jordan algebras and primal-dual interior-point algorithms,” Journal of Comp. and Appl. Math., vol. 86, pp. 149-175, 1997.Google Scholar
  6. 6.
    M.C. Ferris and C. Kanzow, “Complementarity and related problems: A survey,” in P.M. Pardalos and M.G.C. Resende (Eds.), Handbook of Applied Optimization, Oxford University Press: New York, 2002, pp. 514-530.Google Scholar
  7. 7.
    M. Fukushima, Z.Q. Luo, and P. Tseng, “Smoothing functions for second-order-cone complementarity problems,” SIAM Journal on Optimization, vol. 12, pp. 436-460, 2002.Google Scholar
  8. 8.
    Y.R. He and K.F. Ng, “Characterize the boundedness of solution set of generalized complementarity problems,” Math. Prog., in press.Google Scholar
  9. 9.
    Z. Huang, L. Qi, and D. Sun, “Sub-quadratic convergence of a smoothing Newton algorithm for the P0-and monotone LCP,”∼matsundf/.Google Scholar
  10. 10.
    C. Kanzow and C. Nagel, “Semidefinite programs: New search directions, smoothing-type methods, and numerical results,” SIAM Journal on Optimization, vol. 13, pp. 1-23, 2002. http://ifamus.mathematik.∼kanzow/.Google Scholar
  11. 11.
    M.S. Lobo, L. Vandenberghe, S. Boyd, and H. Lebret, “Applications of second-order cone programming,” Linear Alg. Appl., vol. 284, pp. 193-228, 1998.Google Scholar
  12. 12.
    L. McLinden, “Stable monotone variational inequalities,” Math. Prog., vol. 48, pp. 303-338, 1990.Google Scholar
  13. 13.
    J.S. Pang, “Complementarity problems,” In R. Horst and P. Pardalos (Eds.), Handbook in Global Optimization, Kluwer Academic Publishers: Boston, 1994.Google Scholar
  14. 14.
    L. Qi and J. Sun, “A nonsmooth version of Newton's method,” Math. Prog., vol. 58, pp. 353-367, 1993.Google Scholar
  15. 15.
    L. Qi, D. Sun, and G. Zhou, “A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities,” Math. Prog., vol. 87, pp. 1-35, 2001.Google Scholar
  16. 16.
    D. Sun and L. Qi, “Solving variational inequality problems via smoothing-nonsmooth reformulations,” J. Comput. Appl. Math., vol. 129, pp. 37-62, 2001.Google Scholar
  17. 17.
    D. Sun and J. Sun, “Semismooth matrix valued functions,” Math. Oper. Res., vol. 27, pp. 150-169, 2002.Google Scholar
  18. 18.
    J. Sun, D. Sun, and L. Qi, “Quadratic convergence of a squared smoothing Newton method for nonsmooth matrix equations and its applications in semidefinite optimization problems,” Technical Report, Department of Decision Sciences, National University of Singapore 2002, depart/ds/sunjiehomepage/.Google Scholar
  19. 19.
    K.C. Toh, M.J. Todd, and R.H. Tütüncü, “SDPT3-AMatLab software package for semidefinite programming, version 2.1,” Optimization Methods and Software, vol. 11, pp. 545-581, 1999.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • X.D. Chen
    • 1
  • D. Sun
    • 2
  • J. Sun
    • 3
  1. 1.Department of Applied MathematicsTongji UniversityShanghaiChina
  2. 2.Department of MathematicsNational University of SingaporeRepublic of Singapore
  3. 3.SMA and Department of Decision SciencesNational University of SingaporeRepublic of Singapore

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