Journal of Optimization Theory and Applications

, Volume 135, Issue 3, pp 459–473 | Cite as

Conditions for Error Bounds and Bounded Level Sets of Some Merit Functions for the Second-Order Cone Complementarity Problem

  • J.-S. ChenEmail author


Recently this author studied several merit functions systematically for the second-order cone complementarity problem. These merit functions were shown to enjoy some favorable properties, to provide error bounds under the condition of strong monotonicity, and to have bounded level sets under the conditions of monotonicity as well as strict feasibility. In this paper, we weaken the condition of strong monotonicity to the so-called uniform P *-property, which is a new concept recently developed for linear and nonlinear transformations on Euclidean Jordan algebra. Moreover, we replace the monotonicity and strict feasibility by the so-called R 01 or R 02-functions to keep the property of bounded level sets.


Error bounds Jordan products Level sets Merit functions Second-order cones Spectral factorization 


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  1. 1.
    Faraut, U., Korányi, A.: Analysis on Symmetric Cones. Oxford Mathematical Monographs. Oxford University Press, New York (1994) zbMATHGoogle Scholar
  2. 2.
    Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program. 95, 3–51 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Lobo, M.S., Vandenberghe, L., Boyd, S., Lebret, H.: Application of second-order cone programming. Linear Algebra Appl. 284, 193–228 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Alizadeh, F., Schmieta, S.: Symmetric cones, potential reduction methods, and word-by-word extensions. In: Wolkowicz, H., Saigal, R., Vandenberghe, L. (eds.) Handbook of Semidefinite Programming, pp. 195–233. Kluwer Academic, Boston (2000) Google Scholar
  5. 5.
    Andersen, E.D., Roos, C., Terlaky, T.: On implementing a primal-dual interior-point method for conic quadratic optimization. Math. Program. 95, 249–277 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Mittelmann, H.D.: An independent benchmarking of SDP and SOCP solvers. Math. Program. 95, 407–430 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Monteiro, R.D., Tsuchiya, T.: Polynomial convergence of primal-dual algorithms for the second-order cone programs based on the MZ-family of directions. Math. Program. 88, 61–83 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Schmieta, S., Alizadeh, F.: Associative and Jordan algebras, and polynomial time interior-point algorithms for symmetric cones. Math. Oper. Res. 26, 543–564 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Tsuchiya, T.: A convergence analysis of the scaling-invariant primal-dual path-following algorithms for second-order cone programming. Optim. Methods Softw. 11, 141–182 (1999) CrossRefMathSciNetGoogle Scholar
  10. 10.
    Chen, X.-D., Sun, D., Sun, J.: Complementarity functions and numerical experiments for second-order cone complementarity problems. Comput. Optim. Appl. 25, 39–56 (2003) zbMATHCrossRefMathSciNetGoogle 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) CrossRefMathSciNetGoogle Scholar
  12. 12.
    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) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Chen, J.-S.: A new merit function and its related properties for the second-order cone complementarity problem. Pac. J. Optim. 2, 167–179 (2006) zbMATHGoogle Scholar
  14. 14.
    Chen, J.-S.: Two classes of merit functions for the second-order cone complementarity problem. Math. Methods Oper. Res. 64, 495–519 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Chen, J.-S., Tseng, P.: An unconstrained smooth minimization reformulation of the second-order cone complementarity problem. Math. Program. 104, 293–327 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Fischer, A.: A special Newton-type optimization methods. Optimization 24, 269–284 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Fischer, A.: Solution of the monotone complementarity problem with locally Lipschitzian functions. Math. Program. 76, 513–532 (1997) Google Scholar
  18. 18.
    Yamashita, N., Fukushima, M.: A new merit function and a descent method for semidefinite complementarity problems. In: Fukushima, M., Qi, L. (eds.) Reformulation—Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, pp. 405–420. Kluwer Academic, Boston (1999) Google Scholar
  19. 19.
    Luo, Z.-Q., Tseng, P.: A new class of merit functions for the nonlinear complementarity problems. In: Ferris, M.C., Pang, J.-S. (eds.) Complementarity and Variational Problems: State of the Art, pp. 204–225. SIAM, Philadelphia (1997) Google Scholar
  20. 20.
    Tseng, P.: Merit functions for semidefinite complementarity problems. Math. Program. 83, 159–185 (1998) MathSciNetGoogle Scholar
  21. 21.
    Gowda, M.S., Sznajder, R., Tao, J.: Some P-properties for linear transformations on Euclidean Jordan algebras. Linear Algebra Appl. 393, 203–232 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Tao, J., Gowda, M.S.: Some P-properties for the nonlinear transformations on Euclidean Jordan algebra. Math. Oper. Res. 30, 985–1004 (2005) CrossRefMathSciNetGoogle Scholar
  23. 23.
    Sim, C.-K., Zhao, G.: A note on treating a second-order cone program as a special case of a semidefinite program. Math. Program. 102, 609–613 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Liu, Y.-J., Zhang, Z.-W., Wang, Y.-H.: Some properties of a class of merit functions for symmetric cone complementarity problems. Asia-Pac. J. Oper. Res. 23, 473–495 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Malik, M., Mohan, S.R.: On Q and R 0 properties of a quadratic representation in the linear complementarity problems over the second-order cone. Linear Algebra Appl. 397, 85–97 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vols. I and II. Springer, New York (2003) Google Scholar
  27. 27.
    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) CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of MathematicsNational Taiwan Normal UniversityTaipeiTaiwan
  2. 2.Mathematics DivisionNational Center for Theoretical SciencesTaipeiTaiwan

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