  Parallel transmission has been a very promising candidate technology to
mitigate the inevitable radio-frequency field inhomogeneity in magnetic
resonance imaging (MRI) at ultra-high field (UHF). For the first few years,
pulse design utilizing this technique was expressed as a least squares problem
with crude power regularizations aimed at controlling the specific absorption
rate (SAR), hence the patient safety. This approach being suboptimal for many
applications sensitive mostly to the magnitude of the spin excitation, and not
its phase, the magnitude least squares (MLS) problem then was first formulated
in 2007. Despite its importance and the availability of other powerful
numerical optimization methods, this problem yet has been faced exclusively by
the pulse designer with the so-called variable exchange method. In this paper,
we investigate other strategies and incorporate directly the strict SAR and
hardware constraints. Different schemes such as sequential quadratic
programming (SQP), interior point (I-P) methods, semi-definite programming
(SDP) and magnitude squared least squares (MSLS) relaxations are studied both
in the small and large tip angle regimes with real data sets obtained in-vivo
on a human brain at 7 Tesla. Convergence and robustness of the different
approaches are analyzed, and recommendations to tackle this specific problem
are finally given. Small tip angle and inversion pulses are returned in a few
seconds and in under a minute respectively while respecting the constraints,
allowing the use of the proposed approach in routine.
