Prediction-Correction for Nonsmooth Time-Varying Optimization via Forward-Backward Envelopes

TL;DR

This paper introduces a prediction-correction algorithm using forward-backward envelopes for optimizing strongly convex, nonsmooth, time-varying functions, with proven convergence and demonstrated effectiveness in regression tasks.

Contribution

It proposes a novel prediction-correction method leveraging forward-backward envelopes for nonsmooth, time-varying optimization problems, with theoretical convergence guarantees.

Findings

Convergence to a neighborhood of the optimizer depending on sampling time

Effective in time-varying regression with elastic net regularization

Numerical simulations validate the algorithm's performance

Abstract

We present an algorithm for minimizing the sum of a strongly convex time-varying function with a time-invariant, convex, and nonsmooth function. The proposed algorithm employs the prediction-correction scheme alongside the forward-backward envelope, and we are able to prove the convergence of the solutions to a neighborhood of the optimizer that depends on the sampling time. Numerical simulations for a time-varying regression problem with elastic net regularization highlight the effectiveness of the algorithm.