Complementarity Problems Over Symmetric Cones: A Survey of Recent Developments in Several Aspects

Chapter
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 166)

Abstract

The complementarity problem over a symmetric conic (that we call the Symmetric Conic Complementarity Problem, or the SCCP) has received much attention of researchers in the last decade. Many of studies done on the SCCP can be categorized into the three research themes, interior point methods for the SCCP, merit or smoothing function methods for the SCCP, and various properties of the SCCP. In this paper, we will provide a brief survey on the recent developments on these three themes.

References

  1. 1.
    Alizadeh, A., Schmieta, S.H.: Symmetric cones, potential reduction methods and word-by-word extensions. In: Wolkowicz,H., Saigal, R., Vandenberghe, L. (eds.) Handbook of semidefinite programming. Theory, algorithms, and applications. Kluwer Academic Publishers, Boston, Dordrecht, London (2000)Google Scholar
  2. 2.
    Andersen, E., Ye, Y.: On a homogeneous algorithm for the monotone complementarity problems. Math. Program. 84, 375–400 (1999)CrossRefGoogle Scholar
  3. 3.
    Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadelphia, PA (1994)CrossRefGoogle Scholar
  4. 4.
    Chang, Y.L., Pan, S., Chen, J.-S.: Strong semismoothness of Fischer-Burmeister complementarity function associated with symmetric cone. Preprint (2009)Google Scholar
  5. 5.
    Chi, X., Liu, S.: A one-step smoothing Newton method for second-order cone programming. J. Comput. Appl. Math. 223, 114–123 (2009)CrossRefGoogle Scholar
  6. 6.
    Chan, Z.X., Sun, D.: Constraint nondegeneracy, strong regularity, and nonsingularity in semidefinite programming. SIAM J. Optim. 19, 370–396 (2008)CrossRefGoogle Scholar
  7. 7.
    Chen, B., Harkaer, P.T.: A non-interior-point continuation method for linear complementarity problems. SIAM J. Matrix Anal. Appl. 14, 1168–1190 (1993)CrossRefGoogle Scholar
  8. 8.
    Che, B., Harkaer, P.T.: A continuation method for monotone variational inequalities. Math. Program. 69, 237–253 (1995)Google Scholar
  9. 9.
    Chen, C., Mangasarian, O.L.: Smoothing methods for convex inequalities and linear complementarity problems. Math. Program. 71, 51–69 (1995)CrossRefGoogle Scholar
  10. 10.
    Chen, C., Mangasarian, O.L.: A class of smoothing functions for nonlinear and mixed complementarity problems. Comput. Optim. Appl. 5, 97–138 (1996)CrossRefGoogle Scholar
  11. 11.
    Chen, J.-S., Chen, X., Tseng, P.: Analysis of nonsmooth vector-valued functions associated with second-order cones. Math. Program. Ser. B 101, 95–117 (2004)CrossRefGoogle Scholar
  12. 12.
    Chen, J.-S., Tseng, P.: An unconstrained smooth minimization reformulation of the second-order cone complementarity problem. Math. Program. Ser. B 104, 293–327 (2005)CrossRefGoogle Scholar
  13. 13.
    Chen, J.-S.: Two classes of merit functions for the second-order cone complementarity problem. Math. Methods Oper. Res. 64, 495–519 (2006)CrossRefGoogle Scholar
  14. 14.
    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)Google Scholar
  15. 15.
    Chen, J.-S.: Conditions for error bounds and bounded level sets of some merit functions for the second-order cone complementarity problem. J. Optim. Theory Appl. 135, 459–473 (2007)CrossRefGoogle Scholar
  16. 16.
    Chen, J.-S., Pan, S.: A descent method for a reformulation of the second-order cone complementarity problem. J. Comput. Appl. Math. 213, 547–558 (2008)CrossRefGoogle Scholar
  17. 17.
    Chen, X., Tseng, P.: Non-interior continuation methods for solving semidefinite complementarity problems. Math. Program. 95, 431–474 (2003)CrossRefGoogle Scholar
  18. 18.
    Chen, X., Qi, H.-D., Tseng. P: Analysis of nonsmooth symmetric matrix functions with applications to semidefinite complementarity problems. SIAM J. Optim. 13, 960–985 (2003)Google Scholar
  19. 19.
    Chen, X.D., Sun, D., 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)CrossRefGoogle Scholar
  20. 20.
    Chen. X., Qi, H.-D.: Cartesian P-property and its applications to the semidefinite linear complementarity problem. Math. Program. 106, 177–201 (2006)Google Scholar
  21. 21.
    Chua, C.B., Yi, P.: A continuation method for nonlinear complementarity problems over symmetric cone. SIAM J. Optim. 20, 2560–2583 (2010)CrossRefGoogle Scholar
  22. 22.
    Chua, C.B., Lin, H., Yi, P.: Uniform nonsingularity and complementarity problems over symmetric cones. Research report, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore (2009)Google Scholar
  23. 23.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. John Wiley & Sons, New York (1983)Google Scholar
  24. 24.
    Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, New York (1992)Google Scholar
  25. 25.
    Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems Volume I, II. Springer-Verlag, New York, Berlin, Heidelberg (2003)Google Scholar
  26. 26.
    Fang, L., He, G., Huc, Y.: A new smoothing Newton-type method for second-order cone programming problems Appl. Math. Comput. 215, 1020–1029 (2009)Google Scholar
  27. 27.
    Faraut, J., Korányi, A.: Analysis on Symmetric Cones. Oxford Science Publishers, New York (1994)Google Scholar
  28. 28.
    Faybusovich, L.: Euclidean Jordan algebras and interior-point algorithms. Positivity 1, 331–357 (1997)CrossRefGoogle Scholar
  29. 29.
    Fischer, A.: A special Newton-type optimization method. Optim. 24, 269–284 (1992)Google Scholar
  30. 30.
    Fischer, A.: A Newton-type method for positive semidefinite linear complementarity problems. J. Optim. Theory Appl. 86, 585–608 (1995)CrossRefGoogle Scholar
  31. 31.
    Fischer, A.: On the local superlinear convergence of a Newton-type method for LCP under weak conditions. Optim. Methods Software 6, 83–107 (1995)CrossRefGoogle Scholar
  32. 32.
    Fukushima, N., Luo, Z.-Q., Tseng, P.: Smoothing functions for second-order-cone complementarity problems. SIAM J. Optim. 12, 436–460 (2002)CrossRefGoogle Scholar
  33. 33.
    Gowda, M.S., Song, Y.: On semidefinite linear complementarity problems. Math. Program. Ser. A 88, 575–587 (2000)CrossRefGoogle Scholar
  34. 34.
    Gowda, M.S., Song, Y.: Errata: On semidefinite linear complementarity problems. Math. Program. Ser. A 91, 199–200 (2001)Google Scholar
  35. 35.
    Gowda, M.S., Sznajder, R., Tao, J.: Some P-properties for linear transformations on Euclidean Jordan algebras. Special issue on Positivity, Linear Algebra Appl. 393, 203–232 (2004)Google Scholar
  36. 36.
    Gowda, M.S., Tao, J.: Some P-properties for nonlinear transformations on Euclidean Jordan algebras. Math. Oper. Res. 30, 985–1004 (2005)CrossRefGoogle Scholar
  37. 37.
    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)CrossRefGoogle Scholar
  38. 38.
    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)CrossRefGoogle Scholar
  39. 39.
    Gu, J., Zhang, L., Xiao, X.: Log-sigmoid nonlinear Lagrange method for nonlinear optimization problems over second-order cones. J. Comput. Appl. Math. 229, 129–144 (2009)CrossRefGoogle Scholar
  40. 40.
    Gurtuna, F., Petra, C., Potra, F.A., Shevchenko, O., Vancea, A.: Corrector-predictor methods for sufficient linear complementarity problems. Comput Optim Appl. (2009). doi:10.1007/s10589-009-9263-4Google Scholar
  41. 41.
    Han, D.: On the coerciveness of some merit functions for complementarity problems over symmetric cones. J. Math. Anal. Appl. 336, 727–737 (2007)CrossRefGoogle Scholar
  42. 42.
    Hauser, R.A., Güler, O.: Self-scaled barrier functions on symmetric cones and their classification. Found. Comput. Math. 2, 121–143 (2002)Google Scholar
  43. 43.
    Hayashi, S., Yamashita, N., Fukushima, M.: Robust Nash equilibria and second-order cone complementarity problems. J. Nonlinear Convex Anal. 6, 283–296 (2005)Google Scholar
  44. 44.
    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)CrossRefGoogle Scholar
  45. 45.
    Huang, Z.-H., Han, J.: Non-interior continuation method for solving the monotone semidefinite complementarity problem. Appl. Math. Optim. 47, 195–211 (2003)CrossRefGoogle Scholar
  46. 46.
    Huang, Z.-H.: Global Lipschitzian error bounds for semidefinite complementarity problems with emphasis on NCPs. Appl. Math. Comput. 162, 1237–1258 (2005)CrossRefGoogle Scholar
  47. 47.
    Huang, Z.-H., Ni, T.: Smoothing algorithms for complementarity problems over symmetric cones. Comput. Optim. Appl. 45, 557–579 (2010)CrossRefGoogle Scholar
  48. 48.
    Kanno, Y., Martins, J.A.C., Pinto da Costa, A.: Three-dimensional quasi-static frictional contact by using second-order cone linear complementarity problem. Int. J. Numer. Methods Eng. 65, 62–83 (2006)Google Scholar
  49. 49.
    Kanzow, C.: Some noninterior continuation methods for linear complementarity problems. SIAM J. Matrix Anal. Appl. 17, 851–868 (1996)CrossRefGoogle Scholar
  50. 50.
    Kanzow, C.: A new approach to continuation methods for complementarity problems with uniform P-functions. Oper. Res. Lett. 20, 85–92 (1997)CrossRefGoogle Scholar
  51. 51.
    Kanzow, C., Nagel, C.: Quadratic convergence of a nonsmooth Newton-type method for semidefinite programs without strict complementarity. SIAM J. Optim. 15, 654–672 (2005)CrossRefGoogle Scholar
  52. 52.
    Kojima, M., Shindoh, S., Hara, S.: Interior-point methods for the monotone linear complementarity problem in symmetric matrices. SIAM J. Optim. 7, 86–125 (1997)CrossRefGoogle Scholar
  53. 53.
    Kojima, M., Shida, M., Shindoh, S.: Reduction of monotone linear complementarity problems over cones to linear programs over cones. Acta Math. Vietnamica 22, 147–157 (1997)Google Scholar
  54. 54.
    Kojima, M., Shida, M., Shindoh, S.: Local convergence of predictor-corrector infeasible-interior-point method for SDPs and SDLCPs. Math. Program. 80, 129–160 (1998)Google Scholar
  55. 55.
    Kojima, M., Shida, M., Shindoh, S.: A predictor-corrector interior-point algorithm for the semidefinite linear complementarity problem using the Alizadeh-Haeberly-Overton search direction. SIAM J. Optim. 9, 444–465 (1999)CrossRefGoogle Scholar
  56. 56.
    Kong, L., Xiu, N.: New smooth C-functions for symmetric cone complementarity problems. Optim. Lett. 1, 391–400 (2007)CrossRefGoogle Scholar
  57. 57.
    Kong, L., Tunçel, L., Xiu, N.: Monotonicity of Löwner operators and its applications to symmetric cone complementarity problems. Research Report CORR 2007-07, Department of Combinatorics and Optimization, University of Waterloo, Waterloo (2007)Google Scholar
  58. 58.
    Kong, L., Tunçel, L., Xiu, N.: The Fischer-Burmeister complementarity function on Euclidean Jordan algebras. Pac. J. Optim. 6, 423–440 (2010)Google Scholar
  59. 59.
    Kong, L., Xiu, N., Han, J.: The solution set structure of monotone linear complementarity problems over second-order cone. Oper. Res. Lett. 36, 71–76 (2008)CrossRefGoogle Scholar
  60. 60.
    Kong, L., Sun, J., Xiu, N.: A regularized smoothing Newton method for symmetric cone complementarity problems. SIAM J. Optim. 19, 1028–1047 (2008)CrossRefGoogle Scholar
  61. 61.
    Kong, L., Tunçel, L., Xiu, N.: Vector-valued implicit Lagrangian for symmetric cone complementarity problems. Asia-Pac. J. Oper. Res. 26, 199–233 (2009)CrossRefGoogle Scholar
  62. 62.
    Kong, L., Tunçel, L., Xiu, N.: Clarke generalized Jacobian of the projection onto symmetric cones. Set-Valued Variational Anal. 17, 135–151 (2009)CrossRefGoogle Scholar
  63. 63.
    Kong, L., Tunçel, L., Xiu, N.: Homogeneous cone complementarity problems and P properties. Preprint (2009)Google Scholar
  64. 64.
    Korányi, A.: Monotone functions on formally real Jordan algebras. Math. Ann. 269, 73–76 (1984)CrossRefGoogle Scholar
  65. 65.
    Kum, S.H., Lim, Y.D.: Coercivity and strong semismoothness of the penalized Fischer-Burmeister function for the symmetric cone complementarity problem. J. Optim. Theory Appl. 142, 377–383 (2009)CrossRefGoogle Scholar
  66. 66.
    Kum, S.H., Lim, Y.D.: Penalized complementarity functions on symmetric cones. J. Global Optim. 46, 475–485 (2010)CrossRefGoogle Scholar
  67. 67.
    Li, Y.M., Wang, X.T., Wei, D.Y.: A new class of smoothing complementarity functions over symmetric cones. Commun. Nonlinear Sci. Numer. Simul. (2009). doi:10.1016/j.cnsns.2009.12.006Google Scholar
  68. 68.
    Liu, X.-H., Huang, Z.-H.: A smoothing Newton algorithm based on a one-parametric class of smoothing functions for linear programming over symmetric cones. Math. Methods Oper. Res. 70, 385–404 (2009)CrossRefGoogle Scholar
  69. 69.
    Liu, Y., Zhang, L., Liu, M.: Extension of smoothing functions to symmetric cone complementarity problems Appl. Math. J. Chin. Univ. 22, 245–252 (2007)CrossRefGoogle Scholar
  70. 70.
    Löwner, K.: Über monotone matrixfunctionen. Math. Z. 38, 177–216 (1934)Google Scholar
  71. 71.
    Lu, N., Huang, Z.-H., Han, J.: Properties of a class of nonlinear transformations over Euclidean Jordan algebras with applications to complementarity problems. Numer. Funct. Anal. Optim. 30, 799–821 (2009)CrossRefGoogle Scholar
  72. 72.
    Luo, X.-D., Tseng, P.: On a global projection-type error bound for the linear complementarity problem. Linear Algebra Appl. 253, 251–278 (1997)CrossRefGoogle Scholar
  73. 73.
    Lu, Z., Monteiro, R.D.C.: A note on the local convergence of a predictor-corrector interior-point algorithm for the semidefinite linear complementarity problem based on the Alizadeh-Haeberly-Overton search direction. SIAM J. Optim. 15, 1147–1154 (2005)CrossRefGoogle Scholar
  74. 74.
    Luo, Z., Xiu, N.: An O(rL) infeasible interior-point algorithm for symmetric cone LCP via CHKS function. Acta Math. Appl. Sin. (English Series) 25, 593–606 (2009)Google Scholar
  75. 75.
    Luo, Z., Xiu, N.: Path-following interior point algorithms for the Cartesian P  ∗ (κ) LCP over symmetric cones. Sci. China Ser. A: Math. 52, 1769–1784 (2009)CrossRefGoogle Scholar
  76. 76.
    Luo, Z.-Q., Pang, J.-S.: Error bounds for analytic systems and their applications. Math. Program. 67, 1–28 (1995)CrossRefGoogle Scholar
  77. 77.
    Malik, M., Mohan, S.R.: Some geometrical aspects of semidefinite linear complementarity problems. Linear and Multilinear Algebra 54, 55–70 (2006)CrossRefGoogle Scholar
  78. 78.
    Malik, M., Mohan, S.R.: Cone complementarity problems with finite solution sets. Oper. Res. Lett. 34, 121–126 (2006)CrossRefGoogle Scholar
  79. 79.
    Mifflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15, 957–972 (1977)CrossRefGoogle Scholar
  80. 80.
    Mifflin, R.: A modification and an extension of Lemarechal’s algorithm for nonsmooth minimization. Math. Program. Stud. 17, 77–90 (1982)Google Scholar
  81. 81.
    Monteiro, R.D.C, Pang, J.-S.: On two interior-point mappings for nonlinear semidefinite complementarity problems. Math. Oper. Res. 23, 39–60 (1998)Google Scholar
  82. 82.
    Monteiro, R.D.C., Zhang, Y.: A unified analysis for a class of path-following primal-dual interior-point algorithms for semidefinite programming. Math. Program. 81, 281–299 (1998)Google Scholar
  83. 83.
    Muramatsu, M.: On commutative class of search directions for linear programming over symmetric cones. J. Optim. Theory Appl. 112, 595–625 (2002)CrossRefGoogle Scholar
  84. 84.
    Murty, K.G.: On the number of solutions to the complementarity problem and spanning properties of complementary cones. Linear Algebra Appl. 5, 65–108 (1972)CrossRefGoogle Scholar
  85. 85.
    Nesterov, Yu., Nemirovski, A.: Interior-Point Polynomial Algorithms in Convex Programming. SIAM Studies in Applied Mathematics, SIAM, Philadelphia (1994)CrossRefGoogle Scholar
  86. 86.
    Ni, T., Gu, W.-G.: Smoothing Newton algorithm for symmetric cone complementarity problems based on a one-parametric class of smoothing functions. J. Appl. Math. Comput. (2009). doi:10.1007/s12190-009-0341-7Google Scholar
  87. 87.
    Nishimura, R., Hayashi, S., Fukushima, M.: Robust Nash equilibria in N-person non-cooperative games: Uniqueness and reformulation. Pac. J. Optim. 5, 237–259 (2009)Google Scholar
  88. 88.
    Outrata, J.V., Sun, D.: On the coderivative of the projection operator onto the second-order cone. Set-Valued Anal. 16, 999–1014 (2008)CrossRefGoogle Scholar
  89. 89.
    Pan, S., Chen, J.-S.: A one-parametric class of merit functions for the symmetric cone complementarity problem. J. Math. Anal. Appl. 355, 195–215 (2009)CrossRefGoogle Scholar
  90. 90.
    Pan, S., Chen, J.-S.: A regularization method for the second-order cone complementarity problem with the Cartesian P0-property. Nonlinear Anal.: Theory, Methods Appl. 70, 1475–1491 (2009)Google Scholar
  91. 91.
    Pan, S., Chen, J.-S.: A one-parametric class of merit functions for the second-order cone complementarity problem. Comput. Optim. Appl. 45, 581–606 (2010)CrossRefGoogle Scholar
  92. 92.
    Pan, S., Chen, J.-S.: Growth behavior of two classes of merit functions for symmetric cone complementarity problems. J. Optim. Theory Appl. 141, 167–191 (2009)CrossRefGoogle Scholar
  93. 93.
    Pan, S., Chen, J.-S.: A damped Gauss-Newton method for the second-order cone complementarity problem. Appl. Math. Optim. 59, 293–318 (2009)CrossRefGoogle Scholar
  94. 94.
    Pang, J.-S., Gabriel, S.A.: NE/SQP: A robust algorithm for the nonlinear complementarity problem. Math. Program. 60, 295–337 (1993)CrossRefGoogle Scholar
  95. 95.
    Potra, F.A., Sheng, R.: A superlinearly convergent primal-dual infeasible-interior-Point algorithm for semidefinite programming. SIAM J. Optim. 8, 1007–1028 (1998)CrossRefGoogle Scholar
  96. 96.
    Pan, S., Chen, J.-S.: Approximal gradient descent method for the extended second-order cone linear complementarity problem. J. Math. Anal. Appl. 366, 164–180 (2010)CrossRefGoogle Scholar
  97. 97.
    Potra, F.A.: Q-superlinear convergence of the iterates in primal-dual interior-point methods. Math. Program. 91, 99–115 (2001)Google Scholar
  98. 98.
    Potra, F.A.: An infeasible interior point method for linear complementarity problems over symmetric cones. AIP Conf. Proc. 1168, 1403–1406 (2009)CrossRefGoogle Scholar
  99. 99.
    Preiß, M., Stoer, J.: Analysis of infeasible-interior-point paths arising with semidefinite linear complementarity problems. Math. Program. Ser. A 99, 499–520 (2004)CrossRefGoogle Scholar
  100. 100.
    Qin, L., Kong, L., Han, J.: Sufficiency of linear transformations on Euclidean Jordan algebras. Optim. Lett. 3, 265–276 (2009)CrossRefGoogle Scholar
  101. 101.
    Rangarajan, B.K.: Polynomial convergence of infeasible-interior-point methods over symmetric cones. SIAM J. Optim. 16, 1211–1229 (2006)CrossRefGoogle Scholar
  102. 102.
    Renegar, J.: A Mathematical View of Interior Point Methods for Convex Optimization. MPS-SIAM Series on Optimization, SIAM & MPS, Philadelphia (2001)Google Scholar
  103. 103.
    Rui, S.-P., Xu, C.-X.: Inexact non-interior continuation method for solving large-scale monotone SDCP. Appl. Math. Comput. 215, 2521–2527 (2009)CrossRefGoogle Scholar
  104. 104.
    Schmieta, S.H., Alizadeh, F.: Extension of primal-dual interior-point algorithm to symmetric cones. Math. Program. 96, 409–438 (2003)CrossRefGoogle Scholar
  105. 105.
    Shida, M., Shindoh, S., Kojima, M.: Centers of monotone generalized complementarity problems. Math. Oper. Res. 22, 969–976 (1997)CrossRefGoogle Scholar
  106. 106.
    Sim, C.-K., Zhao, G.: Underlying paths in interior point method for monotone semidefinite linear complementarity problem. Math. Program. 110, 475–499 (2007)CrossRefGoogle Scholar
  107. 107.
    Sim, C.-K., Zhao, G.: Asymptotic Behavior of HKM paths in interior point method for monotone semidefinite linear complementarity problem: General theory. J. Optim. Theory Appl. 137, 11–25 (2008)CrossRefGoogle Scholar
  108. 108.
    Sim, C.-K.: On the analyticity of underlying HKM paths for monotone semidefinite linear complementarity problem. J. Optim. Theory Appl. 141, 193–215 (2009)CrossRefGoogle Scholar
  109. 109.
    Sim, C.-K.: Superlinear convergence of an infeasible predictor-corrector path-following interior point algorithm for a semidefinite linear complementarity problem using the Helmberg-Kojima-Monteiro direction. SIAM J. Optim. 21, 102–126 (2011)CrossRefGoogle Scholar
  110. 110.
    Smale, S.: Algorithms for solving equations. In: Gleason, A.M. (ed.) Proceedings of the International Congress of Mathematicians, pp. 172–195. American Mathematical Society, Providence, Rhode Island (1987)Google Scholar
  111. 111.
    Sun, D., Sun, J.: Semismooth matrix valued functions. Math. Oper. Res. 27, 150–169 (2002)CrossRefGoogle Scholar
  112. 112.
    Sun, J., Sun, D., Qi, L.: A squared smoothing Newton method for nonsmooth matrix equations and its applications in semidefinite optimization problems. SIAM J. Optim. 14, 783–806 (2003)CrossRefGoogle Scholar
  113. 113.
    Sun, D., Sun, J.: Strong semismoothness of the Fischer-Burmeister SDC and SOC complementarity functions. Math. Program. 103, 575–581 (2005)CrossRefGoogle Scholar
  114. 114.
    Sun, D., Sun, J.: Löwner’s operator and spectral functions in Euclidean Jordan algebras. Math. Oper. Res. 33, 421–445 (2008)CrossRefGoogle Scholar
  115. 115.
    Sun, J., Zhan, L.: A globally convergent method based on Fischer-Burmeister operators for solving second-order cone constrained variational inequality problems. Comput. Math. Appl. 58, 1936–1946 (2009).CrossRefGoogle Scholar
  116. 116.
    Tao, J., Gowda, M.S.: Some P-properties for nonlinear transformations on Euclidean Jordan algebras. Math. Oper. Res. 30, 985–1004 (2005)CrossRefGoogle Scholar
  117. 117.
    Tao, J.: Positive principal minor property of linear transformations on Euclidean Jordan algebras. J. Optim. Theory Appl. 140, 131–152 (2009)CrossRefGoogle Scholar
  118. 118.
    Tao, J.: Strict semimonotonicity property of linear transformations on Euclidean Jordan algebras. J. Optim. Theory Appl. 144, 575–596 (2010)CrossRefGoogle Scholar
  119. 119.
    Tseng, P.: Merit functions for semi-definite complementarity problems. Math. Program. 83, 159–185 (1998)Google Scholar
  120. 120.
    Tseng, P.: Analysis of a non-interior continuation method based on Chen-Mangasarian smoothing functions for complementarity problems. In: Fukushima, M., Qi, L. (eds.) Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, pp.381–404. Kluwer Academic Publishers, Boston (1998)Google Scholar
  121. 121.
    Xu, S., Burke, J.V.: A polynomial time interior-point path-following algorithm for LCP based on Chen- Harker-Kanzow smoothing techniques. Math. Program. 86, 91–103 (1999)CrossRefGoogle Scholar
  122. 122.
    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 Publishers, Boston (1998)Google Scholar
  123. 123.
    Yan, T., Fukushima, M.: Smoothing method for mathematical programs with symmetric cone complementarity constraints. Technical Report 2009-017, Department of Applied Mathematics and Physics, Kyoto University (2009)Google Scholar
  124. 124.
    Yoshise, A.: Interior point trajectories and a homogeneous model for nonlinear complementarity problems over symmetric cones. SIAM J. Optim. 17, 1129–1153 (2006)CrossRefGoogle Scholar
  125. 125.
    Yoshise, A.: Homogeneous algorithms for monotone complementarity problems over symmetric cones. Pac. J. Optim. 5, 313–337 (2009)Google Scholar
  126. 126.
    Zhang, X., Liu, S., Liu, Z.: A smoothing method for second order cone complementarity problem. J. Comput. Appl. Math. 22, 83–91 (2009)CrossRefGoogle Scholar
  127. 127.
    Yu, Z.: The bounded smooth reformulation and a trust region algorithm for semidefinite complementarity problems. Appl. Math. Comput. 159, 157–170 (2004)CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Graduate School of Systems and Information EngineeringUniversity of TsukubaTsukubaJapan

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