Prediction-correction for Nonsmooth Time-varying Optimization via Forward-backward Envelopes

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.