Minimization of a Large-Scale Quadratic FunctionSubject to a Spherical Constraint

An important problem in linear algebra and optimization is the trust-region subproblem: minimize a quadratic function subject to an ellipsoidal or spherical constraint. This basic problem has several important large-scale applications including seismic inversion and forcing convergence in optimization methods. Existing methods to solve the trust-region subproblem require matrix factorizations, which are not feasible in the large-scale setting. This paper presents an algorithm for solving the large-scale trust-region subproblem that requires a fixed-size limited storage proportional to the order of the quadratic and that relies only on matrix-vector products. The algorithm recasts the trust-region subproblem in terms of a parameterized eigenvalue problem and adjusts the parameter with a superlinearly convergent iteration to find the optimal solution from the eigenvector of the parameterized problem. Only the smallest eigenvalue and corresponding eigenvector of the parameterized problem needs to be computed. The implicitly restarted Lanczos method is well suited to this subproblem.

[1]  J. H. Wilkinson,et al.  AN ESTIMATE FOR THE CONDITION NUMBER OF A MATRIX , 1979 .

[2]  B. Parlett.  The Symmetric Eigenvalue Problem , 1981 .

[3]  W. Gander.  Least squares with a quadratic constraint , 1980 .

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

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

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

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

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

[9]  W. Menke.  Geophysical data analysis : discrete inverse theory , 1984 .

[10]  Richard A. Tapia,et al.  A trust region strategy for nonlinear equality constrained op-timization , 1984 .

[11]  Richard H. Byrd,et al.  A Trust Region Algorithm for Nonlinearly Constrained Optimization , 1987 .

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

[13]  M. El-Alem.  A global convergence theory for the Celis-Dennis-Tapia trust-region algorithm for constrained optimization , 1991 .

[14]  Ya-Xiang Yuan,et al.  A trust region algorithm for equality constrained optimization , 1990, Math. Program..

[15]  Danny C. Sorensen,et al.  Implicit Application of Polynomial Filters in a k-Step Arnoldi Method , 1992, SIAM J. Matrix Anal. Appl..

[16]  Henk A. van der Vorst,et al.  Approximate solutions and eigenvalue bounds from Krylov subspaces , 1995, Numer. Linear Algebra Appl..