  A constrained version of the online convex optimization (OCO) problem is
considered. With slotted time, for each slot, first an action is chosen.
Subsequently the loss function and the constraint violation penalty evaluated
at the chosen action point is revealed. For each slot, both the loss function
as well as the function defining the constraint set is assumed to be smooth and
strongly convex. In addition, once an action is chosen, local information about
a feasible set within a small neighborhood of the current action is also
revealed. An algorithm is allowed to compute at most one gradient at its point
of choice given the described feedback to choose the next action. The goal of
an algorithm is to simultaneously minimize the dynamic regret (loss incurred
compared to the oracle's loss) and the constraint violation penalty (penalty
accrued compared to the oracle's penalty). We propose an algorithm that follows
projected gradient descent over a suitably chosen set around the current
action. We show that both the dynamic regret and the constraint violation is
order-wise bounded by the {\it path-length}, the sum of the distances between
the consecutive optimal actions. Moreover, we show that the derived bounds are
the best possible.
