In this paper we introduce a discrete-time, distributed optimization algorithm executed by a set of agents whose interactions are subject to a communication graph. The algorithm can be applied to optimization costs that are expressed as sums of functions, where each function is associated to an agent. The algorithm can be applied to continuously differentiable cost functions, it is not consensus-based and is derived naturally by solving the first order necessary conditions of a lifted optimization problem with equality constraints. We show that, provided the agents’ initial values are sufficiently closed to a local minimizer and the step-size is sufficiently small, each agent converges to the local minimizer at a linear rate. In addition, we revisit two popular consensus-based distributed optimization algorithms and give sufficient conditions so that there use is extended to non-convex functions as well. We take a closer look at their rate of convergence and also show that unlike our algorithm, for a constant step-size, the consensus-based algorithms do not converge to a local minimizer even though the agents start close enough to the local minimizer.