CLARSTA: A random subspace trust-region algorithm for convex-constrained derivative-free optimization

Abstract

This paper proposes a random subspace trust-region algorithm for general convex-constrained derivative-free optimization (DFO) problems. Similar to previous random subspace DFO methods, the convergence of our algorithm requires a certain accuracy of models and a certain quality of subspaces. For model accuracy, we define a new class of models that is only required to provide reasonable accuracy on the projection of the constraint set onto the subspace. We provide a new geometry measure to make these models easy to analyze, construct, and manage. For subspace quality, we use a normalized random orthogonal projection matrix to provide a method to sample subspaces that preserve the first-order criticality measure by a certain fraction with a probability lower bound independent of the ambient dimension. Based on all these new theoretical results, we present an almost-sure liminf convergence result and an $\mathcal{O}(\epsilon^{-2})$ expected iteration complexity analysis of our algorithm. Numerical experiments on problems with dimensions up to 10000 demonstrate the reliable performance of our algorithm in high dimensions.