MATRO: Metric-Aware Trust-Region Optimization with Fully Quadratic Models

Abstract

Model-based derivative-free trust-region methods build local interpolation models and restrict trial steps to regions where those models are reliable. This paper studies the shape of that region. When an objective is poorly scaled or locally anisotropic, a Euclidean ball can be governed by the steepest local direction and can restrict progress along directions of slow variation. We propose MATRO (Metric-Aware Trust-Region Optimization), a fully quadratic interpolation framework in which the trust region is the ellipsoid s^T M_k s <= Delta_k^2. For any positive definite metric M_k, the induced variable y = M_k^{1/2}s converts the ellipsoidal subproblem into a standard Euclidean trust-region subproblem, so model decrease, ratio tests, radius updates, poisedness, and fully quadratic error bounds can be stated in induced coordinates under a uniform metric contract. The metric is selected from the interpolation Hessian: positive definite quadratics yield a unique volume-normalized curvature metric that isotropizes the induced Hessian and gives a truncated Newton step, while indefinite fitted Hessians motivate an absolute-curvature metric that balances curvature magnitudes without changing curvature signs. Under the standard fully quadratic assumptions and the metric contract, MATRO retains the first-order evaluation-complexity order O(n^2 epsilon^{-2}). Experiments on More-Wild benchmarks, controlled anisotropy tests, and two-dimensional trajectories show that curvature-shaped regions are most effective when the interpolation Hessian captures stable local anisotropy, while dense linear algebra is most visible at loose accuracies or on inexpensive analytic tests.