×

zbMATH — the first resource for mathematics

Convergence of a smoothing algorithm for symmetric cone complementarity problems with a nonmonotone line search. (English) Zbl 1203.90123
Summary: We propose a smoothing algorithm for solving the monotone symmetric cone complementarity problems (SCCP for short) with a nonmonotone line search. We show that the nonmonotone algorithm is globally convergent under an assumption that the solution set of the problem concerned is nonempty. Such an assumption is weaker than those given in most existing algorithms for solving optimization problems over symmetric cones. We also prove that the solution obtained by the algorithm is a maximally complementary solution to the monotone SCCP under some assumptions.

MSC:
90C25 Convex programming
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
Software:
SCCP
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Huang Z H. Sufficient conditions on nonemptiness and boundedness of the solution set of the P 0 function nonlinear complementarity problem. Oper Res Lett, 30: 202–210 (2002) · Zbl 1010.90081
[2] Huang Z H. Locating a maximally complementary solution of the monotone NCP by using non-interior-point smoothing algorithms. Math Methods Oper Res, 61: 41–55 (2005) · Zbl 1066.90127
[3] Zhao Y B, Li D. A globally and locally superlinearly convergent non-interior-point algorithm for P 0 LCPs. SIAM J Optim, 13: 1196–1221 (2003) · Zbl 1039.90081
[4] Zhou G L, Sun D F, Qi L Q. Numerical experiments for a class of squared smoothing Newton methods for box constrained variational inequality problems. In: Fukushima M, Qi L, eds. Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods. Boston: Kluwer Academic Publishers, 1999, 421–441 · Zbl 0928.65080
[5] Gowda M S, Sznajder R. Automorphism invariance of P- and GUS- properties of linear transformations in Euclidean Jordan algebras. Math Oper Res, 31: 109–123 (2006) · Zbl 1168.90620
[6] Gowda M S, Sznajder R, Tao J. Some P-properties for linear transformations on Euclidean Jordan algebras. Linear Algebra Appl, 393: 203–232 (2004) · Zbl 1072.15002
[7] Kong L C, Xiu N H. New smooth C-functions for symmetric cone complementarity problems. Optim Lett, 1(4): 391–400 (2007) · Zbl 1220.90133
[8] Liu Y J, Zhang L 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) · Zbl 1202.90252
[9] Sun D F, Sun J. Löwner’s operator and spectral functions in Euclidean Joedan algebras. Math Oper Res, 33: 421–445 (2008) · Zbl 1218.90197
[10] Yoshise A. Interior point trajectories and a homogeneous model for nonlinear complementarity problems over symmetric cones. SIAM J Optim, 17: 1129–1153 (2006) · Zbl 1136.90039
[11] Faybusovich L. Euclidean Jordan algebra and interior-point algorithm. Positivity, 1: 331–357 (1997) · Zbl 0973.90095
[12] Faybusovich L. Linear systems in Jordan algebras and primal-dual interior-point algorithm. J Comput Appl Math, 86: 149–175 (1997) · Zbl 0889.65066
[13] Schmieta V, Alizadeh F. Associative and Jordan algebras, and polynonial time interior-point algorithms for symmetrc cones. Math Oper Res, 26: 543–564 (2001) · Zbl 1073.90572
[14] Schmieta S, Alizadeh F. Extension of primal-dual interior-point algorithms to symmetric cones. Math Program, 96: 409–438 (2003) · Zbl 1023.90083
[15] Lu Y, Yuan Y X. An interior-point trust region algorithm for general cone progrmming. SIAM J Optim, 18: 65–86 (2007) · Zbl 1145.90079
[16] Liu Y J, Zhang L W, Wang Y H. Analysis of smoothing method for symmetric conic linear programming. J Appl Math Comput, 22: 133–148 (2006) · Zbl 1132.90353
[17] 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) · Zbl 1182.65092
[18] Huang Z H, Ni T. Smoothing algorithms for complementarity problems over symmetric cones. Comput Optim Appl, in press, DOI 10.1007/s10589-008-9180-y, 2008 · Zbl 1198.90373
[19] Kong L C, Tunçel L, Xiu N H. Vector-valued implicit Lagrangian for symmetric con complementarity problems. Asia-Pac J Oper Res, in press · Zbl 1168.90622
[20] Pan S H, Chen J S. A one-parametric class of merit functions for the symmetric cone complementarity problem. J Math Anal Appl, 335: 195–215 (2009) · Zbl 1180.90341
[21] Faraut U, Korányi A. Analysis on Symmetric Cones. Oxford Mathematical Monographs. New York: Oxford University Press, 1994 · Zbl 0841.43002
[22] Karamardian S. Complementarity problems over cones with monotone and pseudomonotone maps. J Optim Theory Appl, 18: 445–454 (1976) · Zbl 0304.49026
[23] Zhang H C, Hager W W. A nonmonotone line search technique and its application to unconstrained optimization. SIAM J Optim, 14: 1043–1056 (2004) · Zbl 1073.90024
[24] Huang Z H, Xu S W. Convergence properties of a non-interior-point smoothing algorithm for the P * NCP. J Ind Manag Optim, 3: 569–584 (2007) · Zbl 1166.90370
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.