  This paper considers unconstrained convex optimization problems with
time-varying objective functions. 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 $1/h$, where $h$ is the length of the sampling interval. The prediction
step is derived by analyzing the iso-residual dynamics of the optimality
conditions. The correction step adjusts for the distance between the current
prediction and the optimizer at each time step, and consists either of one or
multiple gradient steps or Newton 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 $O(h^2)$, and in some cases as
$O(h^4)$, which outperforms the state-of-the-art error bound of $O(h)$ for
correction-only methods in the gradient-correction step. Moreover, when the
characteristics of the objective function variation are not available, we
propose approximate gradient and Newton tracking algorithms (AGT and ANT,
respectively) that still attain these asymptotical error bounds. Numerical
simulations demonstrate the practical utility of the proposed methods and that
they improve upon existing techniques by several orders of magnitude.
