Convergence of trust-region algorithms in metric spaces

TL;DR

This paper extends the convergence analysis of trust-region algorithms to metric spaces, specifically applied to integer optimal control problems with total variation regularization, improving theoretical understanding and computational efficiency.

Contribution

It provides a novel convergence analysis of trust-region methods in metric spaces without trust-region radius resets, applicable to integer optimal control problems.

Findings

Convergence to stationary points is guaranteed without radius resets.

Avoiding radius resets improves computational runtime.

Theoretical analysis aligns with empirical computational improvements.

Abstract

Trust-region algorithms can be applied to very abstract optimization problems because they do not require a specific direction of descent or gradient. This has lead to recent interest in them, in particular in the area of integer optimal control problems, where the infinite-dimensional problem formulations do not assume vector space structure. We analyze a trust-region algorithm in the abstract setting of a metric space, a setting in which integer optimal control problems with total variation regularization can be formulated. Our analysis avoids a reset of the trust-region radius upon acceptance of the iterates when proving convergence to stationary points. This reset has been present in previous analyses of trust-region algorithms for integer optimal control problems. Our computational benchmark shows that the runtime can be considerably improved when avoiding this reset, which is now theoretically justified.