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Efficient Global Optimization (EGO) is a competent evolutionary algorithm suited for problems with limited design parameters and expensive cost functions [1]. Many electromagnetics problems, including some antenna designs, fall into this class, as complex electromagnetics simulations can take substantial computational effort. This makes simple evolutionary algorithms such as genetic algorithms or particle swarms very time-consuming for design optimization, as many iterations of large populations are usually required. When physical experiments are necessary to perform tradeoffs or determine effects which may not be simulated, use of these algorithms is simply not practical at all due to the large numbers of measurements required. In this paper we first present a brief introduction to the EGO algorithm. We then present the parasitic superdirective two-element array design problem and results obtained by applying EGO to obtain the optimal element separation and operating frequency to maximize the array directivity. We compare these results to both the optimal solution and results obtained by performing a similar optimization using the Nelder-Mead downhill simplex method. Our results indicate that, unlike the Nelder-Mead algorithm, the EGO algorithm did not become stuck in local minima but rather found the area of the correct global minimum. However, our implementation did not always drill down into the precise minimum and the addition of a local search technique seems to be indicated.