We investigate the vulnerabilities of consensus-based distributed optimization protocols to nodes that deviate from the prescribed update rule (e.g., due to failures or adversarial attacks). After characterizing certain fundamental limitations on the performance of any distributed optimization algorithm in the presence of adversaries, we propose a robust consensus-based distributed optimization algorithm that is guaranteed to converge to the convex hull of the set of minimizers of the non-adversarial nodes' functions. We also study the distance-to-optimality properties of our proposed robust algorithm in terms of F-local sets of the graph. We show that finding the largest size of such sets is NP-hard.