We present an evolutionary algorithm which predicts stable atomic structures and phase diagrams by searching the energy landscape of empirical and ab initio Hamiltonians. Composition and geometrical degrees of freedom may be varied simultaneously. We show that this method utilizes information from favorable local structure at one composition to predict that at others, achieving far greater efficiency of phase diagram prediction than a method which relies on sampling compositions individually. We detail this and a number of other efficiency-improving techniques implemented in the genetic algorithm for structure prediction code that is now publicly available. We test the efficiency of the software by searching the ternary Zr-Cu-Al system using an empirical embedded-atom model potential. In addition to testing the algorithm, we also evaluate the accuracy of the potential itself. We find that the potential stabilizes several correct ternary phases, while a few of the predicted ground states are unphysical. Our results suggest that genetic algorithm searches can be used to improve the methodology of empirical potential design.