We apply Pareto genetic algorithm (GA) optimization to the design of antenna elements for use in the Long Wavelength Array (LWA), a large, low-frequency radio telescope currently under development. By manipulating antenna geometry, the Pareto GA simultaneously optimizes the received Galactic background or "sky'' noise level and radiation patterns of the antenna over all frequencies. Geometrical constraints are handled explicitly in the GA in order to guarantee the realizability, and to impart control over the monetary cost of the generated designs. The antenna elements considered are broadband planar dipoles arranged horizontally over the ground. It is demonstrated that the Pareto GA approach generates a set of designs, which exhibit a wide range of trade-offs between the two design objectives, and satisfy all constraints. Multiple GA executions are performed to determine how antenna performance trade-offs are affected by different geometrical constraint values, feed impedance values, radiating element shapes and orientations, and ground conditions. Two different planar dipole antenna designs are constructed, and antenna input impedance and sky noise drift scan measurements are performed to validate the results of the GA.