Analysis and Synthesis of Switched Optimization Algorithms

TL;DR

This paper develops a framework for analyzing and synthesizing discrete-time optimization algorithms that maintain exponential convergence rates despite network-induced delays and switching dynamics, ensuring robustness in communication-constrained environments.

Contribution

It introduces a method using linear matrix inequalities and Zames-Falb filters to certify and synthesize robust optimization algorithms over dynamic networks.

Findings

Algorithms achieve exponential convergence under time-varying delays.

Robustness is demonstrated over networks with unstable channel dynamics.

The approach effectively certifies convergence rates via LMIs and filter design.

Abstract

Deployment of optimization algorithms over communication networks face challenges associated with time delays and corruptions. Fixed time delays can destabilize popular gradient-based algorithms, and this degradation is exacerbated by time-varying delays that may arise from packet drops. This work concentrates on the analysis and synthesis of discrete-time optimization algorithms with certified exponential convergence rates that are robust against switched network dynamics between the optimizer and the gradient oracle. Analysis is accomplished by solving linear matrix inequalities under bisection in the exponential convergence rate, searching over Zames-Falb filter coefficients that can certify convergence. Synthesis is performed by alternating between a search over filter coefficient for a fixed controller, and a search over controllers for a fixed filter. Effectiveness is demonstrated by the synthesis of convergent optimization algorithms over networks with time-varying delays, and networks with unstable channel dynamics.