The goal of optimal power flow (OPF) is to find a minimum cost production of committed generating units while satisfying technical constraints of the power system. To ensure robustness of the network, the system must be able to find new operating points within the technical limits in the event of component failures such as line and generator outages. However, finding an optimal, or even a feasible, preventive/corrective action may be difficult due to the innate nonconvexity of the problem. With the goal of finding a global solution to the post-contingency OPF problem of a stressed network, e.g. a network with a line outage, we apply a homotopy method to the problem. By parametrizing the constraint set, we define a series of optimization problems to represent a gradual outage and iteratively solve these problems using local search. Under the condition that the global minimum of the OPF problem for the base-case is attainable, we find theoretical guarantees to ensure that the OPF problem for the contingency scenario will also converge to its global minimum. We show that this convergence is dependent on the geometry of the homotopy path. The effectiveness of the proposed approach is demonstrated on Polish networks.