In this paper we introduce two discrete-time, distributed optimization algorithms executed by a set of agents whose interactions are subject to a communication graph. The algorithms can be applied to optimization problems where the cost function is expressed as a sum of functions, and where each function is associated to an agent. In addition, the agents can have equality constraints as well. The algorithms are not consensus-based and can be applied to non-convex optimization problems with equality constraints. We demonstrate that the first distributed algorithm results naturally from applying a first order method to solve the first order necessary conditions of a lifted optimization problem with equality constraints; optimization problem whose solution embeds the solution of our original problem. We show that, provided the agents’ initial values are sufficiently close to a local minimizer, and the step-size is sufficiently small, under standard conditions on the cost and constraint functions, each agent converges to the local minimizer at a linear rate. Next, we use an augmented Lagrangian idea to derive a second distributed algorithm whose local convergence requires weaker sufficient conditions than in the case of the first algorithm.