A social network is an example of a phenomenon with complex stochastic dependence that is commonly modeled with a class of exponential families called exponential random graph models (ERGM). Maximum likelihood estimators (MLE) for such exponential families can be difficult to estimate when the likelihood is difficult to compute. Most methodologies rely on iterated estimates and are sensitive to the starting value, failing to converge if started too far from the solution. Even more problematic is that the MLE may not exist, a situation that occurs with positive probability for ERGMs. In such a case, the MLE is actually "at infinity" in some direction of the parameter space.
Here we present a simple line search algorithm to find the MLE of a regular exponential family when the MLE exists and is unique. The algorithm can be started from any initial value and avoids trial-and-error experimentation. When the MLE does not exist, our algorithm adapts Geyer's (2009a) approach to detect non-existent MLEs and construct one-sided confidence intervals for how close the parameter is to infinity.