“…5,6 Several robust optimization methods have been proposed and have been shown to have great advantages by achieving robust plans while maintaining high plan quality. 2,[7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] Unfortunately, in proton therapy besides uncertainties, there are machine hardware constraints, for example, field-size constraint, minimum-maximum energy constraint, dose-rate constraint for synchrotron-based proton therapy systems, and minimum-maximum monitor unit (MU) constraint. Among them, the minimum MU constraint is the most prominent one.…”