We propose an algorithm for localizing multiple failures in an IP/WDM network. They can be either hard failures (unexpected events that interrupt suddenly the established channels) or soft failures (events that progressively degrade the quality of transmission). Hard failures are detected at the WDM layer, whereas soft failures can be detected at the optical layer if proper testing equipment is deployed, and/or by performance monitoring at a higher layer, which is here IP. The algorithm also tolerates missing and false alarms. Even without missing and false alarms, multiple fault localization is NP-hard. The diagnosis phase (i.e., the localization of the faulty components upon reception of the alarms) can however remain very fast, but at the expense of a very complex precomputation phase, carried out whenever the optical channels are set up or cleared down. We show how the algorithm performs on an example of an IP/WDM network.