Primal and Dual Prediction-Correction Methods for Time-Varying Convex Optimization

We propose a unified framework for time-varying convex optimization based on the prediction-correction paradigm, both in the primal and dual spaces. In this framework, a continuously varying optimization problem is sampled at fixed intervals, and each problem is approximately solved with a primal or dual correction step. The solution method is warm-started with the output of a prediction step, which solves an approximation of a future problem using past information. Prediction approaches are studied and compared under different sets of assumptions. Examples of algorithms covered by this framework are time-varying versions of the gradient method, splitting methods, and the celebrated alternating direction method of multipliers (ADMM).