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

TL;DR

This paper introduces CLARSTA, a novel random subspace trust-region method for convex-constrained derivative-free optimization, with theoretical convergence guarantees and successful high-dimensional numerical results.

Contribution

It develops new models and geometric measures for subspace sampling, enabling reliable convergence analysis and high-dimensional optimization performance.

Findings

Algorithm converges globally with high probability.

Numerical tests show effectiveness up to 10,000 dimensions.

New geometric measure simplifies model analysis and construction.

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 the concentration of measure on the Grassmann manifold to provide a method to sample subspaces that preserve the first-order criticality measure by a certain fraction with a certain probability lower bound. Based on all these new theoretical results, we present an almost-sure global convergence and a worst-case complexity analysis of our algorithm. Numerical experiments on problems with dimensions up to 10000 demonstrate the reliable performance of our algorithm in high dimensions.