Analyzing the Effect of Persistent Asset Switches on a Class of Hybrid-Inspired Optimization Algorithms

TL;DR

This paper investigates how persistent switching of objectives affects continuous-time convex optimization algorithms, extending stability analysis to hybrid systems and differential inclusions, with implications for robustness and efficiency.

Contribution

It extends stability results for switched systems to differential inclusions in optimization, providing convergence characterization under switching constraints.

Findings

Convergence to Omega-limit sets characterized for switched systems.

Semi-global practical stability established under dwell-time constraints.

Results applicable to hybrid and differential inclusion-based optimization algorithms.

Abstract

Convex optimization challenges are currently pervasive in many science and engineering domains. In many applications of convex optimization, such as those involving multi-agent systems and resource allocation, the objective function can persistently switch during the execution of an optimization algorithm. Motivated by such applications, we analyze the effect of persistently switching objectives in continuous-time optimization algorithms. In particular, we take advantage of existing robust stability results for switched systems with distinct equilibria and extend these results to systems described by differential inclusions, making the results applicable to recent optimization algorithms that employ differential inclusions for improving efficiency and/or robustness. Within the framework of hybrid systems theory, we provide an accurate characterization, in terms of Omega-limit sets, of the set to which the optimization dynamics converge. Finally, by considering the switching signal to be constrained in its average dwell time, we establish semi-global practical asymptotic stability of these sets with respect to the dwell-time parameter.