Abstract
In this paper, we consider smooth convex approximations to the maximum eigenvalue function. To make it applicable to a wide class of applications, the study is conducted on the composite function of the maximum eigenvalue function and a linear operator mapping ℝm to \({\mathcal{S}}_n \), the space of n-by-n symmetric matrices. The composite function in turn is the natural objective function of minimizing the maximum eigenvalue function over an affine space in \({\mathcal{S}}_n \). This leads to a sequence of smooth convex minimization problems governed by a smoothing parameter. As the parameter goes to zero, the original problem is recovered. We then develop a computable Hessian formula of the smooth convex functions, matrix representation of the Hessian, and study the regularity conditions which guarantee the nonsingularity of the Hessian matrices. The study on the well-posedness of the smooth convex function leads to a regularization method which is globally convergent.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Auslender, A. (1999), Penalty and barrier methods: a unified framework, SIAM J. Optimization 10, 211-230.
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.
Bertsekas, D.P. (1982), Constrained Optimization and Lagrangian Multiplier Methods, Academic Press, New York.
Chang, P.-L. (1980), A minimax approach t nonlinear programming, Doctoral Dissertation, University of Washington, Department of Mathematics.
Chen, X. and Tseng, P. (2003), Non-interior continuation methods for solving semidefinite complementarity problems, Mathematical Programming 95, 431-474.
Dontchev, A.L. and Zolezzi, T. (1993), Well-posed optimization problems. In: Lecture Notes in Mathematics, 1543. Springer, Berlin.
Facchinei, F. (1998), Structural and stability properties of P 0 nonlinear complementarity problems, Mathematics of Operations Research 23, 735-745.
Facchinei, F. and Kanzow, C. (1999), Beyond monotonicity in regularization methods for nonlinear complementarity problems, SIAM J. Control and Optimization 37, 1150-1161.
Goldstein, A.A. (1997), Chebyshev approximation and linear inequalities via exponentials, Report, Department of Mathematics, University of Washington, Seattle.
Golub, G.H. and Val Loan, C.F. (1996), Matrix Computations, 3rd ed., Johns Hopkins University Press, Baltimore, MD.
Horn, R.A. and Johnson, C.R. (1985), Matrix Analysis, Cambridge Press, Cambridge, United Kingdom.
Lewis, A.S. (1996), Derivatives of spectral functions, Mathematics of Operations Research 21, 576-588.
Lewis, A.S. and Overton, M.L. (1996), Eigenvalue optimization, Acta Numerica 5, 149-190.
Lewis, A.S. and Sendov, H.S. (2001), Twice differentiable spectral functions, SIAM J. Matrix Analysis and Application 23, 368-386.
Li, X.-S. (1991), An aggregation function method for nonlinear programming, Science in China (Series A) 34, 1467-1473.
Oustry, F. (1999), The U-Lagrangian of the maximum eigenvalue function, SIAM J. Optimization 9, 526-549.
Oustry, F. (2000), A second-order bundle method to minimize the maximum eigenvalue function, Mathematical Programming 89, 1-33.
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.
Peng, J.-M. and Lin, Z.-H. (1999), A non-interior continuation method for generalized linear complementarity problems, Mathematical Programming 86, 533-563.
Qi, H.-D. (1999), Tikhonov regularization methods for variational inequality problems, J. Optimization Theory and Application 102, 193-201.
Qi, H.-D. (2000), A regularized smoothing Newton method for box constrained variational inequality problems with P 0-functions, SIAM J. Optimization 10, 315-330.
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.
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.
Ravindran, G. and Gowda, M.S. (2000), Regularization of P 0-functions in box variational inequality problems, SIAM J. Optimization 11, 748-760.
Shapiro, A. and Fan, M.K.H. (1995), On eigenvalue optimization, SIAM J. Optimization 5, 552-568.
Stewart, G.W. and Sun, J.-G. (1990), Matrix Perturbation Theory, Academic Press, New York.
Sun, D. (1990), A regularization Newton method for solving nonlinear complementarity problems, Applied Mathematics and Optimization 40, 315-339.
Tang, H. and Zhang, L. (1994), A maximum entropy method for convex programming, Chinese Science Bulletin 39, 682-684.
Tseng, P. and Bertsekas, D.P. (1993), On the convergence f the exponential multiplier method for convex programming, Mathematical Programming 60, 1-19.
Zhang, L. and Tang, H. (1998), A further study on a penalty function of Bertsekas. Advances in nonlinear programming (Beijing, 1996), Appl. Optim., 14, Kluwer Academic Publishers, Dordrecht, pp. 345-351.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, X., Qi, H., Qi, L. et al. Smooth Convex Approximation to the Maximum Eigenvalue Function. J Glob Optim 30, 253–270 (2004). https://doi.org/10.1007/s10898-004-8271-2
Issue Date:
DOI: https://doi.org/10.1007/s10898-004-8271-2