BUP-TR: Bayesian Underdetermined Projection Trust-Region Methods for Derivative-Free Optimization

Abstract

Underdetermined quadratic interpolation is a central model-construction tool in model-based derivative-free trust-region methods: it limits sampling costs but leaves an affine family of interpolating quadratics. Classical solvers select one element of this family by prescribing a fixed norm or model-change measure, such as the least-Frobenius-change Hessian update in Powell-type methods. We introduce BUP-TR (Bayesian Underdetermined Projection Trust-Region), which instead completes the model by projecting a prior quadratic onto the affine interpolation set in the precision norm supplied by the prior. The same precision matrix defines a spectral geometry certificate, MAP-poisedness, and a repair mechanism for interpolation sets. Under standard smoothness assumptions, uniform precision bounds, MAP-poisedness, and a trust-region-scale prior-accuracy condition, the hard-MAP models are fully linear. Consequently, BUP-TR attains global first-order convergence and O(epsilon^{-2}) evaluation complexity, with geometry-repair evaluations included. A NEWUOA-style implementation, BUP-NEWUOA, improves fixed-budget performance on the reported benchmark suite at moderate and stringent accuracy targets while retaining the computational structure of a Powell-type trust-region method.