The Analysis of Optimization Algorithms, A Dissipativity Approach

TL;DR

This paper applies dissipativity theory from control systems to analyze and improve the convergence of iterative optimization algorithms, providing a systematic, control-inspired framework for their selection and tuning.

Contribution

It formalizes the connection between optimization algorithms and control systems using dissipativity theory, enabling systematic analysis and tuning of algorithms.

Findings

Dissipativity theory can characterize convergence properties of optimization algorithms.

Control-theoretic framework allows systematic selection and tuning of algorithms.

Examples demonstrate the effectiveness of the dissipativity approach.

Abstract

Optimization problems in engineering and applied mathematics are typically solved in an iterative fashion, by systematically adjusting the variables of interest until an adequate solution is found. The iterative algorithms that govern these systematic adjustments can be viewed as a control system. In control systems, the output in measured and the input is adjusted using feedback to drive the error to zero. Similarly, in iterative algorithms, the optimization objective is evaluated and the candidate solution is adjusted to drive it toward the optimal point. Choosing an algorithm that works well for a variety of optimization problems is akin to robust controller design. Just as dissipativity theory can be used to analyze the stability properties of control systems, it can also be used to analyze the convergence properties of iterative algorithms. By defining an appropriate notion of "energy" that dissipates with every iteration of the algorithm, the convergence properties of the algorithm can be characterized. This article formalizes the connection between iterative algorithms and control systems and shows through examples how dissipativity theory can be used to analyze the performance of many classes of optimization algorithms. This control-theoretic viewpoint enables the selection and tuning of optimization algorithms to be performed in an automated and systematic way.