Solving Trust Region Subproblems Using Riemannian Optimization

TL;DR

This paper introduces Riemannian optimization algorithms to solve a variant of the Trust Region Subproblem with equality constraints, providing global convergence insights and preconditioning strategies for improved performance.

Contribution

It develops a family of Riemannian algorithms for the constrained Trust Region Subproblem, linking it to eigenvector problems and analyzing convergence with preconditioning.

Findings

Algorithms converge to a global optimum.

Deep connection between Trust Region Subproblem and eigenvector problems.

Preconditioning improves convergence rates.

Abstract

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.