Simultaneous optimization of radio frequency and gradient waveforms with exact Hessians and slew rate constraints applied to k-points excitation.

We consider an excitation pulse with piecewise constant gradient trajectories and radio frequency (RF) waveforms such that the solution of the Bloch equations without relaxation terms can be represented by rotations. Based on this analytic solution we formulate a non-linear program for finding sub-pulse durations, gradient strengths, and complex RF voltages which minimize the deviation between the achieved and desired magnetization. We develop explicit expressions for the first and second order derivatives of the objective function. We extend the non-linear program to precisely account for gradient slew rate constraints. Using an interior point solver we apply the developed theory to simultaneously optimize the positions of k-points, their associated RF voltages and durations.