  In this paper, we propose a self-triggered algorithm to solve a class of
convex optimization problems with time-varying objective functions. It is known
that the trajectory of the optimal solution can be asymptotically tracked by a
continuous-time state update law. Unfortunately, implementing this requires
continuous evaluation of the gradient and the inverse Hessian of the objective
function which is not amenable to digital implementation. Alternatively, we
draw inspiration from self-triggered control to propose a strategy that
autonomously adapts the times at which it makes computations about the
objective function, yielding a piece-wise affine state update law. The
algorithm does so by predicting the temporal evolution of the gradient using
known upper bounds on higher order derivatives of the objective function. Our
proposed method guarantees convergence to arbitrarily small neighborhood of the
optimal trajectory in finite time and without incurring Zeno behavior. We
illustrate our framework with numerical simulations.
