We have developed the ALAMO methodology with the aim of producing a tool capable of using a small number of data points for learning algebraic models that are accurate and as simple as possible. This approach relies on (a) integer nonlinear optimization to build low-complexity models from input-output data, (b) derivative-free optimization to systematically improve the models, and (c) global optimization to enforce physical constraints and insights. We present computational results and comparisons between ALAMO and a variety of learning techniques, including Latin hypercube sampling, simple least-squares regression, and the lasso. We also describe results from applications in chemical process design that motivated the development of ALAMO.