  This paper considers a class of convex optimization problems where both, the
objective function and the constraints, have a continuously varying dependence
on time. Our goal is to develop an algorithm to track the optimal solution as
it continuously changes over time inside or on the boundary of the dynamic
feasible set. We develop an interior point method that asymptotically succeeds
in tracking this optimal point in nonstationary settings. The method utilizes a
time varying constraint slack and a prediction-correction structure that relies
on time derivatives of functions and constraints and Newton steps in the
spatial domain. Error free tracking is guaranteed under customary assumptions
on the optimization problems and time differentiability of objective and
constraints. The effectiveness of the method is illustrated in a problem that
involves multiple agents tracking multiple targets.
