Dual Prediction-Correction Methods for Linearly Constrained Time-Varying Convex Programs

TL;DR

This paper introduces dual prediction-correction algorithms for linearly constrained, time-varying convex programs, achieving improved tracking accuracy over classical methods by leveraging dual space optimization.

Contribution

It extends prediction-correction methods to linearly constrained problems in the dual space, providing similar error bounds as primal approaches.

Findings

Achieves $h^2$ order asymptotic tracking error.

Extends prediction-correction to linearly constrained problems.

Demonstrates effectiveness in dual space for time-varying convex programs.

Abstract

Devising efficient algorithms to solve continuously-varying strongly convex optimization programs is key in many applications, from control systems to signal processing and machine learning. In this context, solving means to find and track the optimizer trajectory of the continuously-varying convex optimization program. Recently, a novel prediction-correction methodology has been put forward to set up iterative algorithms that sample the continuously-varying optimization program at discrete time steps and perform a limited amount of computations to correct their approximate optimizer with the new sampled problem and predict how the optimizer will change at the next time step. Prediction-correction algorithms have been shown to outperform more classical strategies, i.e., correction-only methods. Typically, prediction-correction methods have asymptotic tracking errors of the order of $h^2$, where $h$ is the sampling period, whereas classical strategies have order of $h$. Up to now, Prediction-correction algorithms have been developed in the primal space, both for unconstrained and simply constrained convex programs. In this paper, we show how to tackle linearly constrained continuously-varying problem by prediction-correction in the dual space and we prove similar asymptotic error bounds as their primal versions.