Strong semismoothness of the Fischer-Burmeister SDC and SOC complementarity functions. (English) Zbl 1099.90062

Summary: We show that the Fischer-Burmeister complementarity functions, associated to the semidefinite cone (SDC) and the second order cone (SOC), respectively, are strongly semismooth everywhere. Interestingly enough, the proof relys on a relationship between the singular value decomposition of a nonsymmetric matrix and the spectral decomposition of a symmetric matrix.


90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
90C22 Semidefinite programming
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65F18 Numerical solutions to inverse eigenvalue problems
Full Text: DOI


[1] Bonnans, J.F., Cominetti, R., Shapiro, A.: Second order optimality conditions based on parabolic second order tangent sets. SIAM J. Optim. 9, 466–493 (1999) · Zbl 0990.90127
[2] Borwein, J.M., Lewis, A.S.: Convex Analysis and Nonlinear Optimization: Theory and Examples. Springer-Verlag, New York, 2000 · Zbl 0953.90001
[3] Chen, J., Chen, X., Tseng, P.: Analysis of nonsmooth vector-valued functions associated with second-order cones. Math. Prog. Series B 101, 95–117 (2004) · Zbl 1065.49013
[4] Chen, X., Qi, H., Tseng, P.: Analysis of nonsmooth symmetric-matrix functions with applications to semidefinite complementarity problems. SIAM J. Optim. 13, 960–985 (2003) · Zbl 1076.90042
[5] Chen, X.,Tseng, P.: Non-interior continuation methods for solving semidefinite complementarity problems. Math. Prog. 95, 431–474 (2003) · Zbl 1023.90046
[6] 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) · Zbl 1038.90084
[7] Chu, M.T.: Inverse eigenvalue problems. SIAM Rev. 40, 1–39 (1998) · Zbl 0915.15008
[8] Fischer, A.: A special Newton-type optimization method. Optimization 24, 269–284 (1992) · Zbl 0814.65063
[9] Fischer, A.: Solution of monotone complementarity problems with locally Lipschitzian functions. Math. Prog. Series B 76, 513–532 (1997) · Zbl 0871.90097
[10] Fukushima, M., Luo, Z.Q., Tseng, P.: Smoothing functions for second-order-cone complementarity problems. SIAM J. Optim. 12, 436–460 (2002) · Zbl 0995.90094
[11] Golub, G.H., Van Loan, C.F.: Matrix Computations. 3rd edn, The Johns Hopkins University Press, Baltimore, USA, 1996 · Zbl 0865.65009
[12] Kanzow, C., Nagel, C.: Semidefinite programs: new search directions, smoothing-type methods. SIAM J. Optim. 13, 1–23 (2002) · Zbl 1029.90052
[13] Pang, J.-S., Qi, L.: Nonsmooth equations: motivation and algorithms. SIAM J. Optim. 3, 443–465 (1993) · Zbl 0784.90082
[14] Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Prog. 58, 353–367 (1993) · Zbl 0780.90090
[15] Sun, D., Sun, J.: Semismooth matrix valued functions. Math. Oper. Res. 27, 150–169 (2002) · Zbl 1082.49501
[16] Sun, D., Sun, J.: Strong semismoothness of eigenvalues of symmetric matrices and its application to inverse eigenvalue problems. SIAM J. Numer. Anal. 40, 2352–2367 (2002) · Zbl 1041.65037
[17] 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 (2004) · Zbl 1079.90094
[18] Tseng, P.: Merit functions for semidefinite complementarity problems. Math. Prog. 83, 159–185 (1998) · Zbl 0920.90135
[19] Yamashita, N., Fukushima, M.: A new merit function and a descent method for semidefinite complementarity problems. In: M. Fukushima, L. Qi (eds.), Reformulation - Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Boston, Kluwer Academic Publishers, 1999, pp. 405–420 · Zbl 0969.90087
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.