Smooth convex approximation to the maximum eigenvalue function

Journal of Global Optimization

Volumn 30, Issue 2, 2004, Pages 253-270

  Chen, Xin ^a   Qi, Houduo ^c   Qi, Liqun ^b   Teo, Kok Lay ^b  

^a UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

^b HONG KONG POLYTECHNIC UNIVERSITY   (Hong Kong)

^c UNIVERSITY OF SOUTHAMPTON   (United Kingdom)

Author keywords

Matrix representation; Spectral function; Symmetric function; Tikhonov regularization

Indexed keywords

MATRIX REPRESENTATION; SPECTRAL FUNCTIONS; SYMMETRIC FUNCTION; TIKHONOV REGULARIZATION;

APPROXIMATION THEORY; CONVERGENCE OF NUMERICAL METHODS; EIGENVALUES AND EIGENFUNCTIONS; FUNCTIONS; OPTIMIZATION; PARAMETER ESTIMATION;

MATRIX ALGEBRA;

EID: 14644435032     PISSN: 09255001     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10898-004-8271-2     Document Type: Conference Paper

References (30)

1. Auslender, A. (1999), Penalty and barrier methods: a unified framework, SIAM J. Optimization 10, 211-230.
2. Ben-Tal, A. and Teboulle, M. (1989), A smoothing technique for nondifferentiable optimization problems. In: Optimization, 1-11. Fifth French German Conference, Lecture Notes in Math. 1405, Springer, New York.
3. Bertsekas, D.P. (1982), Constrained Optimization and Lagrangian Multiplier Methods, Academic Press, New York.
4. Chang, P.-L. (1980), A minimax approach to nonlinear programming, Doctoral Dissertation, University of Washington, Department of Mathematics.
5. Chen, X. and Tseng, P. (2003), Non-interior continuation methods for solving semidefinite complementarity problems, Mathematical Programming 95, 431-474.
6. Dontchev, A.L. and Zolezzi, T. (1993), Well-posed optimization problems. In: Lecture Notes in Mathematics, 1543. Springer, Berlin.
7. 0 nonlinear complementarity problems, Mathematics of Operations Research 23, 735-745.
8. Facchinei, F. and Kanzow, C. (1999), Beyond monotonicity in regularization methods for nonlinear complementarity problems, SIAM J. Control and Optimization 37, 1150-1161.
9. Goldstein, A.A. (1997), Chebyshev approximation and linear inequalities via exponentials, Report, Department of Mathematics, University of Washington, Seattle.
10. Golub, G.H. and Val Loan, C.F. (1996), Matrix Computations, 3rd ed., Johns Hopkins University Press, Baltimore, MD.
11. Horn, R.A. and Johnson, C.R. (1985), Matrix Analysis, Cambridge Press, Cambridge, United Kingdom.
12. Lewis, A.S. (1996), Derivatives of spectral functions, Mathematics of Operations Research 21, 576-588.
13. Lewis, A.S. and Overton, M.L. (1996), Eigenvalue optimization, Acta Numerica 5, 149-190.
14. Lewis, A.S. and Sendov, H.S. (2001), Twice differentiable spectral functions, SIAM J. Matrix Analysis and Application 23, 368-386.
15. Li, X.-S. (1991), An aggregation function method for nonlinear programming, Science in China (Series A) 34, 1467-1473.
16. Oustry, F. ( 1999), The script U sign-Lagrangian of the maximum eigenvalue function, SIAM J. Optimization 9, 526-549.
17. Oustry, F. (2000), A second-order bundle method to minimize the maximum eigenvalue function, Mathematical Programming 89, 1-33.
18. Overton, M.L. and Womersley, R.S. (1995), Second derivatives for optimizing eigenvalues of symmetric matrices, SIAM J. Matrix Analysis and Application 16, 667-718.
19. Peng, J.-M. and Lin, Z.-H. (1999), A non-interior continuation method for generalized linear complementarity problems, Mathematical Programming 86, 533-563.
20. Qi, H.-D. (1999), Tikhonov regularization methods for variational inequality problems, J. Optimization Theory and Application 102, 193-201.
21. 0-functions, SIAM J. Optimization 10, 315-330.
22. Qi, H.-D. and Liao, L.-Z. (1999), A smoothing Newton method for extended vertical linear complementarity problems, SIAM J. Matrix Analysis and Application 21, 45-66.
23. Qi L. and Tseng, P. (2002), An analysis of piecewise smooth functions and almost smooth functions, Department of Applied Mathematics, The Hong Kong Polytechnic University.
24. 0-functions in box variational inequality problems, SIAM J. Optimization 11, 748-760.
25. Shapiro, A. and Fan, M.K.H. (1995), On eigenvalue optimization, SIAM J. Optimization 5, 552-568.
26. Stewart, G.W. and Sun, J.-G. (1990), Matrix Perturbation Theory, Academic Press, New York.
27. Sun, D. (1990), A regularization Newton method for solving nonlinear complementarity problems, Applied Mathematics and Optimization 40, 315-339.
28. Tang, H. and Zhang, L. (1994), A maximum entropy method for convex programming, Chinese Science Bulletin 39, 682-684.
29. Tseng, P. and Bertsekas, D.P. (1993), On the convergence of the exponential multiplier method for convex programming, Mathematical Programming 60, 1-19.
30. Zhang, L. and Tang, H. (1998), A further study on a penalty function of Bertsekas. Advances in non-linear programming (Beijing, 1996), Appl. Optim., 14, Kluwer Academic Publishers, Dordrecht, pp. 345-351.