The optimization, in the minimax sense, of an endfire array antenna with constraints on side lobes over a certain region and the Q-factor is presented for variable amplitude and phase excitation. The technique employed consists of the combination of an augmented Lagrangian algorithm and a direct search method of optimization. The problem with constraints on only side lobes, and the one with constraints on side lobes and the Q-factor have been solved for a 0.4 λ spaced linear array of 10 isotropic elements, and the amplitude and phase required to achieve optimal directivity are determined. These results are illustrated and compared with the optimum case without any constraint and the one with constraint on only the Q-factor. Also, the optimal directivity is shown for various specified side lobe levels.