A feedback control approach to convex optimization with inequality constraints

TL;DR

This paper introduces a continuous-time feedback control algorithm for inequality-constrained convex optimization, offering faster convergence and simpler analysis compared to primal-dual methods.

Contribution

It presents a novel control-inspired algorithm that improves convergence speed and simplifies theoretical analysis for convex optimization with inequalities.

Findings

The proposed method converges faster than primal-dual gradient dynamics.

Exponential convergence is proven for smooth, strongly convex problems.

The algorithm's convergence rate is more straightforward to assess.

Abstract

We propose a novel continuous-time algorithm for inequality-constrained convex optimization inspired by proportional-integral control. Unlike the popular primal-dual gradient dynamics, our method includes a proportional term to control the primal variable through the Lagrange multipliers. This approach has both theoretical and practical advantages. On the one hand, it simplifies the proof of the exponential convergence in the case of smooth, strongly convex problems, with a more straightforward assessment of the convergence rate concerning prior literature. On the other hand, through several examples, we show that the proposed algorithm converges faster than primal-dual gradient dynamics. This paper aims to illustrate these points by thoroughly analyzing the algorithm convergence and discussing some numerical simulations.