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Computational Optimization and Applications

, Volume 45, Issue 3, pp 581–606 | Cite as

A one-parametric class of merit functions for the second-order cone complementarity problem

  • Jein-Shan Chen
  • Shaohua Pan
Article

Abstract

We investigate a one-parametric class of merit functions for the second-order cone complementarity problem (SOCCP) which is closely related to the popular Fischer–Burmeister (FB) merit function and natural residual merit function. In fact, it will reduce to the FB merit function if the involved parameter τ equals 2, whereas as τ tends to zero, its limit will become a multiple of the natural residual merit function. In this paper, we show that this class of merit functions enjoys several favorable properties as the FB merit function holds, for example, the smoothness. These properties play an important role in the reformulation method of an unconstrained minimization or a nonsmooth system of equations for the SOCCP. Numerical results are reported for some convex second-order cone programs (SOCPs) by solving the unconstrained minimization reformulation of the KKT optimality conditions, which indicate that the FB merit function is not the best. For the sparse linear SOCPs, the merit function corresponding to τ=2.5 or 3 works better than the FB merit function, whereas for the dense convex SOCPs, the merit function with τ=0.1, 0.5 or 1.0 seems to have better numerical performance.

Keywords

Second-order cone Complementarity Merit function Jordan product 

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References

  1. 1.
    Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program. 95, 3–51 (2003) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    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
  3. 3.
    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) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chen, J.-S., Tseng, P.: An unconstrained smooth minimization reformulation of the second-order cone complementarity problem. Math. Program. 104, 293–327 (2005) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chen, J.-S., Chen, X., Tseng, P.: Analysis of nonsmooth vector-valued functions associated with second-order cone. Math. Program. 101, 95–117 (2004) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    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) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vols. I and II. Springer, New York (2003) Google Scholar
  8. 8.
    Ferris, M.C., Mangasarian, O.L., Pang, J.-S. (eds.): Complementarity: Applications, Algorithms and Extensions. Kluwer Academic, Dordrecht (2001) MATHGoogle Scholar
  9. 9.
    Fukushima, M., Luo, Z.-Q., Tseng, P.: Smoothing functions for second-order cone complementarity problems. SIAM J. Optim. 12, 436–460 (2002) CrossRefMathSciNetGoogle Scholar
  10. 10.
    Grippo, L., Lampariello, F., Lucidi, S.: A nonmonotone line search technique for Newton’s method. SIAM J. Numer. Anal. 23, 707–716 (1986) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    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) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kanzow, C., Kleinmichel, H.: A new class of semismooth Newton-type methods for nonlinear complementarity problems. Comput. Optim. Appl. 11, 227–251 (1998) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Lobo, M.S., Vandenberghe, L., Boyd, S., Lebret, H.: Application of second-order cone programming. Linear Algebra Appl. 284, 193–228 (1998) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Luo, Z.-Q., Tseng, P.: A new class of merit functions for the nonlinear complementarity problem. In: Ferris, M.C., Pang, J.-S. (eds.) Complementarity and Variational Problems: State of the Art, pp. 204–225. SIAM, Philadelphia (1997) Google Scholar
  15. 15.
    Mittelmann, H.D.: An independent benchmarking of SDP and SOCP solvers. Math. Program. 95, 407–430 (2003) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Monteiro, R.D.C., 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) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (1999) MATHCrossRefGoogle Scholar
  18. 18.
    Pataki, G., Schmieta, S.: The DIMACS library of semidefinite-quadratic-linear programs. Preliminary draft, Computational Optimization Research Center, Columbia University, New York. http://dimacs.rutgers.edu/Challenges
  19. 19.
    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

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of MathematicsNational Taiwan Normal UniversityTaipeiTaiwan
  2. 2.Mathematics DivisionNational Center for Theoretical Sciences, Taipei OfficeTaipeiTaiwain
  3. 3.School of Mathematical SciencesSouth China University of TechnologyGuangzhouChina

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