# Prediction-Correction Splittings for Nonsmooth Time-Varying Optimization

## TL;DR

This paper introduces a prediction-correction splitting algorithm for solving nonsmooth, time-varying convex optimization problems, with proven convergence and demonstrated effectiveness in robotic formation control.

## Contribution

It develops a novel online prediction-correction framework using splitting methods for nonsmooth, time-varying optimization, with convergence guarantees.

## Key findings

- Convergence to a neighborhood of the optimal solution trajectory.
- Effective performance demonstrated in robotic leader-following simulations.
- Applicable to a broad class of nonsmooth, time-varying problems.

## Abstract

We address the solution of time-varying optimization problems characterized by the sum of a time-varying strongly convex function and a time-invariant nonsmooth convex function. We design an online algorithmic framework based on prediction-correction, which employs splitting methods to solve the sampled instances of the time-varying problem. We describe the prediction-correction scheme and two splitting methods, the forward-backward and the Douglas-Rachford. Then by using a result for generalized equations, we prove convergence of the generated sequence of approximate optimizers to a neighborhood of the optimal solution trajectory. Simulation results for a leader following formation in robotics assess the performance of the proposed algorithm.