This manuscript describes a method of gradient waveform design by nonlinear constrained optimization. Methods of formulation and solution of the waveform optimization problem are briefly described for minimization of root mean squared current and minimization of waveform moments. Waveforms generated using these objectives are presented and compared with those obtained with other objectives. The method uses waveforms which are defined as a set of discrete amplitudes in order to remove artificial constraints on waveform shape imposed by "multilobe" designs. These point-to-point amplitudes are the parameters determined in the optimization procedure which includes knowledge of the specific imaging conditions and the specific gradient hardware system. Some beneficial results of this design approach are: a) physically realizable waveforms which optimally achieve specific imaging and motion artifact reduction goals, b) waveforms which are guaranteed to be optimal with respect to one of several possible objective, c) less reliance on the experience of the designer, and d) a potential reduction in waveform design time.