TY - JOUR
T1 - Solving trust region subproblems using Riemannian optimization
AU - Mor, Uria
AU - Shustin, Boris
AU - Avron, Haim
N1 - Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2023/6
Y1 - 2023/6
N2 - The Trust Region Subproblem is a fundamental optimization problem that takes a pivotal role in Trust Region Methods. However, the problem, and variants of it, also arise in quite a few other applications. In this article, we present a family of iterative Riemannian optimization algorithms for a variant of the Trust Region Subproblem that replaces the inequality constraint with an equality constraint, and converge to a global optimum. Our approach uses either a trivial or a non-trivial Riemannian geometry of the search-space, and requires only minimal spectral information about the quadratic component of the objective function. We further show how the theory of Riemannian optimization promotes a deeper understanding of the Trust Region Subproblem and its difficulties, e.g., a deep connection between the Trust Region Subproblem and the problem of finding affine eigenvectors, and a new examination of the so-called hard case in light of the condition number of the Riemannian Hessian operator at a global optimum. Finally, we propose to incorporate preconditioning via a careful selection of a variable Riemannian metric, and establish bounds on the asymptotic convergence rate in terms of how well the preconditioner approximates the input matrix.
AB - The Trust Region Subproblem is a fundamental optimization problem that takes a pivotal role in Trust Region Methods. However, the problem, and variants of it, also arise in quite a few other applications. In this article, we present a family of iterative Riemannian optimization algorithms for a variant of the Trust Region Subproblem that replaces the inequality constraint with an equality constraint, and converge to a global optimum. Our approach uses either a trivial or a non-trivial Riemannian geometry of the search-space, and requires only minimal spectral information about the quadratic component of the objective function. We further show how the theory of Riemannian optimization promotes a deeper understanding of the Trust Region Subproblem and its difficulties, e.g., a deep connection between the Trust Region Subproblem and the problem of finding affine eigenvectors, and a new examination of the so-called hard case in light of the condition number of the Riemannian Hessian operator at a global optimum. Finally, we propose to incorporate preconditioning via a careful selection of a variable Riemannian metric, and establish bounds on the asymptotic convergence rate in terms of how well the preconditioner approximates the input matrix.
UR - http://www.scopus.com/inward/record.url?scp=85163170128&partnerID=8YFLogxK
U2 - 10.1007/s00211-023-01360-0
DO - 10.1007/s00211-023-01360-0
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AN - SCOPUS:85163170128
SN - 0029-599X
VL - 154
SP - 1
EP - 33
JO - Numerische Mathematik
JF - Numerische Mathematik
IS - 1-2
ER -