Abstract
This paper focuses on the study of differential properties of the symmetric matrix-valued Fischer–Burmeister (FB) function. As the main results, the formulas for the directional derivative, the B-subdifferential and the generalized Jacobian of the symmetric matrix-valued Fischer–Burmeister function are established, which can be utilized in designing implementable Newton-type algorithms for nonsmooth equations involving the symmetric matrix-valued FB function.
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Sun, D.F., Sun, J.: Strong semismoothness of the Fischer-Burmeister SDC and SOC complementarity functions. Math. Program. 103, 575–582 (2005)
Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993)
Sim, C.K., Sun, J., Ralph, D.: A note on the Lipschitz continuity of the gradient of the squared norm of the matrix-valued Fischer-Burmeister function. Math. Program. 107, 547–553 (2006)
Ding, C., Sun, D.F., Toh, K.C.: An introduction to a class of matrix cone programming, report. Department of mathematics. National University of Singapore, Singapore (2010)
Stewart, G.W., Sun, J.: Matrix Perturbation Theory. Academic Press, New York (1990)
Fischer, A.: A special Newton-type optimization method. Optimization 24, 269–284 (1992)
Tseng, P.: Merit functions for semidefinite complementarity problems. Math. Program. 83, 159–185 (1998)
Kanzow, C., Nagel, C.: Semidefinite programs: new search directions, smoothing-type methods. SIAM J. Optim. 13, 1–23 (2002)
Bi, S.J., Pan, S.H., Chen, J.S.: Nonsingularity conditions for FB system of nonlinear SDPs. SIAM J. Optim. (2012) (to appear)
Torki, M.: Second-order directional derivatives of all eigenvalues of a symmetric matrix. Nonlinear Anal. 46, 1133–1150 (2001)
Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)
Chen, X., Qi, H.D., Tseng, P.: Analysis of nonsmooth symmetric-matrix-valued functions with applications to semidefinite complement problems. SIAM J. Optim. 13, 960–985 (2003)
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Communicated by Liqun Qi.
The authors are grateful to the anonymous referees for their helpful suggestions and comments and thank Professor Shaohua Pan from South China University of Technology for helping us to prove Proposition 4.1.
The work is Supported by the National Natural Science Foundation of China under projects No. 11071029 and No. 11171049 and the Fundamental Research Funds for the Central Universities.
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Zhang, L., Zhang, N. & Pang, L. Differential Properties of the Symmetric Matrix-Valued Fischer-Burmeister Function. J Optim Theory Appl 153, 436–460 (2012). https://doi.org/10.1007/s10957-011-9962-8
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DOI: https://doi.org/10.1007/s10957-011-9962-8