A Class of Prediction-Correction Methods for Time-Varying Convex Optimization

TL;DR

This paper introduces prediction-correction algorithms for tracking solutions of time-varying convex optimization problems, achieving higher accuracy than existing methods through novel analysis and practical algorithms.

Contribution

It develops a new class of prediction-correction methods with improved error bounds for time-varying convex optimization, including approximate algorithms when problem characteristics are unknown.

Findings

Asymptotic error bounds of O(h^2) and O(h^4) for the proposed methods.

Proposed algorithms outperform existing correction-only techniques by several orders of magnitude.

Numerical simulations validate the effectiveness and practical utility of the methods.

Abstract

This paper considers unconstrained convex optimization problems with time-varying objective functions. We propose algorithms with a discrete time-sampling scheme to find and track the solution trajectory based on prediction and correction steps, while sampling the problem data at a constant rate of $1/h$, where $h$ is the length of the sampling interval. The prediction step is derived by analyzing the iso-residual dynamics of the optimality conditions. The correction step adjusts for the distance between the current prediction and the optimizer at each time step, and consists either of one or multiple gradient steps or Newton steps, which respectively correspond to the gradient trajectory tracking (GTT) or Newton trajectory tracking (NTT) algorithms. Under suitable conditions, we establish that the asymptotic error incurred by both proposed methods behaves as $O(h^2)$, and in some cases as $O(h^4)$, which outperforms the state-of-the-art error bound of $O(h)$ for correction-only methods in the gradient-correction step. Moreover, when the characteristics of the objective function variation are not available, we propose approximate gradient and Newton tracking algorithms (AGT and ANT, respectively) that still attain these asymptotical error bounds. Numerical simulations demonstrate the practical utility of the proposed methods and that they improve upon existing techniques by several orders of magnitude.