An approach towards beamforming for a uniform linear array (ULA) based on a novel optimization algorithm, designated as Fibonacci branch search (FBS) is presented in this paper. The proposed FBS search strategy was inspired from Fibonacci sequence principle and uses a fundamental branch structure and interactive searching rules to obtain the global optimal solution in the search space. The structure of FBS is established by two types of multidimensional points on the basis of shortening fraction formed by the Fibonacci sequence, and in this mode, interactive global searching and local optimization rules are implemented alternately to reach global optima, avoiding stagnating in local optimum. At the same time, the rigorous mathematical proof for the accessibility and convergence of FBS towards the global optimum is presented to further verify the validity of our theory and support our claim. Taking advantage of the global search ability and high convergence rate of this technique, a robust adaptive beamformer technique is also constructed here by FBS as a real time implementation to improve the beamforming performance by preventing the loss of optimal trajectory. The performance of the FBS is compared with five typical heuristic optimization algorithms, and the reported simulation results demonstrate the superiority of the proposed FBS algorithm in locating the optimal solution with higher precision and reveal the further improvement in adaptive beamforming performance.