The techniques of linear programming have been applied to the synthesis of optimum directional patterns from arrays of point elements. The simplex method has been used to generate optimum element excitations for a given array when the desired directional characteristics are specified. Various criteria are included in the definition of optimality. Numerical examples are given for arrays of up to 70 elements. The size and geometry of arrays that can be treated by this approach are limited only by computation time and computer storage requirements.