This paper studies a class of distributed optimization problems, whose objective is to minimize the global cost function formed by a sum of local cost functions through local information exchanges. For undirected connected graphs, we propose two distributed optimization algorithms based on the proportional-integral feedback mechanism. Under the condition that the local cost functions are differentiable and convex, it is proved that the proposed algorithms asymptotically converge to a global minimum. For the case that the local cost functions have local Lipschitz gradient and the global cost function is strongly convex with respect to the global minimum, the exponential convergences of the two distributed optimization algorithms are established. In addition, in order to avoid continuous communication between agents and reduce communication burden, by integrating the two proposed distributed optimization algorithms with event-triggered communications, two event-triggered based distributed optimization algorithms are developed. It is shown that the two proposed event-triggered optimization algorithms are free of Zeno behavior. Moreover, the two proposed event-triggered based distributed optimization algorithms maintain the same convergence properties as the distributed optimization algorithms with continuous communications under the corresponding conditions. Finally, the above theoretical results are verified by numerical simulations. Copyright