First-order perturbation theory of trust-region subproblems

TL;DR

This paper develops first-order perturbation theory for the trust-region subproblem, providing conditions for stability, defining condition numbers, and demonstrating that solutions can be well-conditioned even in nearly hard cases, with practical implications.

Contribution

It introduces the first-order perturbation analysis for TRS, including conditions for easy case stability, condition number definitions, and insights into solution conditioning in nearly hard cases.

Findings

Perturbation bounds are sharp and reliable.

Solutions can be well-conditioned even in nearly hard cases.

The analysis helps evaluate ill-conditioning of TRS before solving.

Abstract

Trust-region subproblem (TRS) is an important problem arising in many applications such as numerical optimization, Tikhonov regularization of ill-posed problems, and constrained eigenvalue problems. In recent decades, extensive works focus on how to solve the trust-region subproblem efficiently. To the best of our knowledge, there are few results on perturbation analysis of the trust-region subproblem. In order to fill in this gap, we focus on first-order perturbation theory of the trust-region subproblem. The main contributions of this paper are three-fold. First, suppose that the TRS is in \emph{easy case}, we give a sufficient condition under which the perturbed TRS is still in easy case. Second, with the help of the structure of the TRS and the classical eigenproblem perturbation theory, we perform first-order perturbation analysis on the Lagrange multiplier and the solution of the TRS, and define their condition numbers. Third, we point out that the solution and the Lagrange multiplier could be well-conditioned even if TRS is in {\it nearly hard case}. The established results are computable, and are helpful to evaluate ill-conditioning of the TRS problem beforehand. Numerical experiments show the sharpness of the established bounds and the effectiveness of the proposed strategies.