A simple and practical adaptive trust-region method

Abstract

We present an adaptive trust-region method for unconstrained optimization that allows inexact solutions to the trust-region subproblems. Our method is a simple variant of the classical trust-region method of Ssorensen. The method achieves the best possible convergence bound up to an additive logarithmic term for finding an $ε$-approximate stationary point, i.e., $O( Δ_f L^{1/2} ε^{-3/2}) + \tilde{O}(1)$ iterations, where $L$ is the Lipschitz constant of the Hessian, $Δ_f$ is the optimality gap, and $ε$ is the termination tolerance for the gradient norm. This improves over existing trust-region methods whose worst-case bound is at least a factor of $L$ worse. We compare our performance with state-of-the-art trust-region (TRU) and cubic regularization (ARC) methods from the GALAHAD library on the CUTEst benchmark problems with at least 100 variables. We also compare with the recently developed Universal trust-region (UTR) method, using the same subproblem solver. In terms of shifted geometric mean of wall-clock times our method's is between $1.2\times$ and $2\times$ faster. We report similar improvements for number of function evaluations, factorizations, gradient evaluations and Hessian evaluations. Compared to the conference version of this paper, our revised method includes several practical enhancements. These modifications dramatically improved performance, including almost an order of magnitude reduction in the shifted geometric mean of wall-clock times. We also show that it suffices for the function to be continuously twice-differentiable to guarantee that either the minimum gradient norm converges to zero or the objective value tends towards negative infinity, even when the iterates diverge.