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This paper explores the problem of avoiding local minima solutions in entropy-based synthetic aperture radar (SAR) autofocus. These autofocus algorithms correct defocused SAR images by determining the phase error estimate that produces the image with minimum entropy. However, the optimization strategy may converge to local minima solutions that correspond to incorrect image restorations. We propose two methods for reducing the likelihood of achieving such solutions. The first is a novel wavelet-based decomposition technique that determines the neighborhood of the global entropy minimum. A second strategy is the application of simulated annealing techniques to the optimization. We explore the performance of these methods using simulated SAR data, and provide a justification for how they work. Worst case phase errors in which the phase is random and uncorrelated between elements are considered.