Gauss Quadrature Applied to Trust Region Computations

This paper discusses the application of the Lanczos process to the solution of large-scale trust-region subproblems. Techniques pioneered by Golub based on Gauss quadrature are applied to derive inexpensively computable upper and lower bounds for quantities of interest. These bounds help determine how many steps of the Lanczos process should be carried out.

[1]  G. Golub,et al.  Estimation of the L-Curve via Lanczos Bidiagonalization , 1999 .

[2]  Danny C. Sorensen,et al.  A New Matrix-Free Algorithm for the Large-Scale Trust-Region Subproblem , 2000, SIAM J. Optim..

[3]  David M. author-Gay.  Computing Optimal Locally Constrained Steps , 1981 .

[4]  William W. Hager,et al.  Minimizing a Quadratic Over a Sphere , 2001, SIAM J. Optim..

[5]  Philippe L. Toint,et al.  Towards an efficient sparsity exploiting newton method for minimization , 1981 .

[6]  Nicholas I. M. Gould,et al.  Trust Region Methods , 2000, MOS-SIAM Series on Optimization.

[7]  T. Steihaug.  The Conjugate Gradient Method and Trust Regions in Large Scale Optimization , 1983 .

[8]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[9]  Ya-Xiang Yuan,et al.  On the truncated conjugate gradient method , 2000, Math. Program..

[10]  Nicholas I. M. Gould,et al.  Solving the Trust-Region Subproblem using the Lanczos Method , 1999, SIAM J. Optim..

[11]  Gene H. Golub,et al.  Matrix computations , 1983 .

[12]  Jorge J. Moré,et al.  Computing a Trust Region Step , 1983 .

[13]  Carl Tim Kelley,et al.  Iterative methods for optimization , 1999, Frontiers in applied mathematics.

[14]  Jorge J. Moré,et al.  Recent Developments in Algorithms and Software for Trust Region Methods , 1982, ISMP.

[15]  Martin Hanke,et al.  A Note on Tikhonov Regularization of Large Linear Problems , 2003 .

[16]  D. Sorensen.  Newton's method with a model trust region modification , 1982 .

[17]  G. Golub,et al.  Quadratically constrained least squares and quadratic problems , 1991 .