## A New Matrix-Free Algorithm for the Large-Scale Trust-Region Subproblem (1995)

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@MISC{Santos95anew,

author = {Sandra A. Santos and Danny C. Sorensen},

title = {A New Matrix-Free Algorithm for the Large-Scale Trust-Region Subproblem},

year = {1995}

}

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### Abstract

The trust-region subproblem arises frequently in linear algebra and optimization applications. Recently, matrix-free methods have been introduced to solve large- scale trust-region subproblems. These methods only require a matrix-vector product and do not rely on matrix factorizations [4, 7]. These approaches recast the trust- region subproblem in terms of a parameterized eigenvalue problem and then adjust the parameter to find the optimal solution from the eigenvector corresponding to the smallest eigenvalue of the parameterized eigenvalue problem. This paper presents a new matrix-free algorithm for the large-scale trust-region subproblem. The new algorithm improves upon the previous algorithms by introducing a unified iteration that naturally includes the so called hard case. The new iteration is shown to be superlinearly convergent in all cases. Computational results are presented to illustrate convergence properties and robustness of the method.