Iterative thresholding methods have been extensively studied as faster alternatives to convex optimization methods for solving large-sized problems in compressed sensing. A novel iterative thresholding method called LCAMP (Location Constrained Approximate Message Passing) is presented for reducing computational complexity and improving reconstruction accuracy when a nonzero location (or sparse support) constraint can be obtained from view shared images. LCAMP modifies the existing approximate message passing algorithm by replacing the thresholding stage with a location constraint, which avoids adjusting regularization parameters or thresholding levels. This work is first compared with other conventional reconstruction methods using random one-dimention signals and then applied to dynamic contrast-enhanced breast magnetic resonance imaging to demonstrate the excellent reconstruction accuracy (less than 2% absolute difference) and low computation time (5–10 s using Matlab) with highly undersampled three-dimentional data (244 × 128 × 48; overall reduction factor = 10). Magn Reson Med 70:370–381, 2013. © 2012 Wiley Periodicals, Inc.