This article addresses the distributed optimization problem in the presence of malicious adversaries that can move within the network and induce faulty behaviors in the attacked nodes. We first investigate the vulnerabilities of a consensus-based secure distributed optimization protocol under mobile adversaries. Then, a modified resilient distributed optimization algorithm is proposed. We develop conditions on the network structure for both complete and non-complete directed graph cases, under which the proposed algorithm guarantees that the estimates by regular nodes converge to the convex combination of the minimizers of their local functions. Simulations are carried out to verify the effectiveness of our approach.