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Synthesis of array patterns involving optimization with linear and nonlinear constraints is considered. The general solution for the composite weight vector can be decomposed into a part which satisfies the linear constraints and another lying orthogonal to it. This decomposition is shown to yield simple solutions to several beam optimization problems including some with nonlinear constraints. A geometrical derivation of this result is presented which also provides new insights into the mechanism of null-steering via linear constraints. Application of this approach in conjunction with some search algorithms to the synthesis of optimum array patterns with prescribed narrow or broad nulls is shown to yield interesting and useful solutions. The results are illustrated by considering the design of "circular" and "arc" arrays.