Consider a connected network of agents endowed with local cost functions representing private objectives. Agents seek to find an agreement on some minimizer of the aggregate cost, by means of repeated communications between neighbors. This paper investigates the case where some agents are unreliable in the sense that they permanently inject some false value in the network. We introduce a new relaxation of the initial optimization problem. We show that the relaxed problem is equivalent to the initial one under some regularity conditions which are characterized. We propose two iterative distributed algorithms for finding minimizers of the relaxed problem. When all agents are reliable, these algorithms converge to the sought consensus provided that the above regularity conditions are satisfied. In the presence of misbehaving agents, we show in simple scenario that our algorithms converge to a solution which remains in the vicinity of the sought consensus. Unlike standard distributed algorithms, our approach turns out to be unsensitive to large perturbations. Numerical experiments complete our theoretical results