Prediction-correction methods for time-varying convex optimization

We consider unconstrained convex optimization problems with objective functions that vary continuously in time. 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. The prediction step is derived by analyzing the iso-residual dynamics of the optimality conditions, while the correction step consists either of one or multiple gradient steps or Newton's 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(h2), and in some cases as O(h4), which outperforms the state-of-the-art error bound of O(h) for correction-only methods in the gradient-correction step. Numerical simulations demonstrate the practical utility of the proposed methods.