Distributed approaches have many computational benefits, but they are vulnerable to attacks from a subset of devices transmitting incorrect information. This paper investigates Byzantine-resilient algorithms in a decentralized setting, where devices communicate directly with one another. We leverage the so-called dual approach to design a general robust decentralized optimization method. We provide both global and local clipping rules in the special case of average consensus, with tight convergence guarantees. These clipping rules are practical, and yield results that finely characterize the impact of Byzantine nodes, highlighting for instance a qualitative difference in convergence between global and local clipping thresholds. Lastly, we demonstrate that they can serve as a basis for designing efficient attacks.