A magnitude-least-squares radiofrequency pulse design algorithm is reported which uses interleaved exact and stochastically-generated inexact updates to escape local minima and find low-cost solutions. Inexact updates are performed using a small randomly selected minibatch of the available B1+ measurements to update RF pulse weights, which perturbs the sequence of alternating projections. Applications to RF shimming, parallel transmit spokes RF pulse design, and spectral-spatial RF pulse design are considered. Numerical and simulation studies characterized the optimal minibatch size, which was found to consistently produce lower power and lower RMSE solutions across subjects, coil geometries, B1+ resolutions and orientations. The method was validated in-vivo at 7 Tesla and produced improvements in image quality in a slice-by-slice RF-shimmed imaging sequence. Compared to conventional methods, the pulse design method can more robustly design RF pulses that correct for B1+ inhomogeneities at ultra-high field strengths, and enable pulse designs to be completed with increased computational efficiency