On Convergence of Binary Trust-Region Steepest Descent

TL;DR

This paper improves the convergence analysis of binary trust-region steepest descent (BTR), showing it converges to stationary points and exhibits a regularizing effect, supported by computational validation and hybridization with CIA.

Contribution

The paper provides enhanced convergence results for BTR under a compactness assumption, linking it to CIA and demonstrating its regularizing properties.

Findings

BTR converges weakly-* to stationary points of the relaxed problem.

BTR has a regularizing effect on solutions.

Hybridization of CIA and BTR improves solution quality.

Abstract

Binary trust-region steepest descent (BTR) and combinatorial integral approximation (CIA) are two recently investigated approaches for the solution of optimization problems with distributed binary-/discrete-valued variables (control functions). We show improved convergence results for BTR by imposing a compactness assumption that is similar to the convergence theory of CIA. As a corollary we conclude that BTR also constitutes a descent algorithm on the continuous relaxation and its iterates converge weakly-$^*$ to stationary points of the latter. We provide computational results that validate our findings. In addition, we observe a regularizing effect of BTR, which we explore by means of a hybridization of CIA and BTR.