Focusing on the testable implications on the equilibrium manifold, we show that the rationalizability problem is NP-complete. Subsequently, we present an integer programming (IP) approach to characterizing general equilibrium models. This approach avoids the use of the Tarski-Seidenberg algorithm for quantifier elimination that is commonly used in the literature. The IP approach naturally applies to settings with any number of observations, which is attractive for empirical applications. In addition, it can easily be adjusted to analyze the testable implications of alternative general equilibrium models (that include, e.g., public goods, externalities and/or production). Further, we show that the IP framework can easily address recoverability questions (pertaining to the structural model that underlies the observed equilibrium behavior), and account for empirical issues when bringing the IP methodology to the data (such as goodness-of-fit and power). Finally, we show how to develop easy-to-implement heuristics that give a quick (but possibly inconclusive) answer to whether or not the data satisfy the general equilibrium models.