In this paper we consider a discretized version of the problem of optimal beam-forming, or radar transmit and receive pattern design, for stationary radar target localization in the presence of white Gaussian noise. We assume that the target is equally likely to be in one of N discrete cells and the number of allowed observations L is strictly less than N, making an exhaustive search not feasible. We propose two new approaches for beam-form design in target localization problems: a fixed, off-line beam-form design approach and an adaptive, on-line beam-form design technique. The beam-form is designed off-line in the fixed approach to minimize the probability of error after exactly L observations. In particular, the decision is available only after the last (Lthe) observation is acquired. We show that this fixed beam-form design approach is directly related to signal constellations design in digital communications. By contrast, the beam-form is optimized after each observation to minimize the probability of incorrectly localizing the target after the next observation is acquired at each step of the process. The optimization relies on the previously acquired information. The adaptive approach has a better performance than the fixed one. Unlike binary search, these two approaches can work with any number of observations. This work falls under the area of optimal search, which deals with optimal allocation of effort in search problems. The need for optimal search strategies arises in many areas such as the radar target localization problems that we address here, fault location in circuits, localization of mobile stations in wireless networks and Internet information searches.