N-1-1 contingency analysis considers the consecutive loss of two elements in a power system, with intervening time for operator adjustments; the associated reliability criterion was recently included in the NERC Standard TPL-001-1. In this paper, we introduce optimization models for N-1-1 contingency analysis, based on DC optimal power flow considerations. We use mixed-integer programming approaches to optimally model the system adjustments required to avoid potential cascading outages during the primary and secondary contingencies. Contingencies are determined via worst-case interdiction analysis. To facilitate operation during the secondary contingency, line overloads and load shedding are allowed. We test our models and algorithms on several IEEE test systems. Our computational experiments indicate potential for the models to augment comprehensive system operations models, such as unit commitment.