Supplementary Information:Impact of Non-Hermiticity and Nonlinear Interactions on Disorder-Induced Localized Modes

Section A of the Supporting In- formation provides a detailed description of the iterative op- timization method. Section B presents additional examples showing the impact of gain saturation on the spatial profile of the modes. Section C illustrates the results of FDTD numeri- cal simulations, highlighting spectral stability at higher pump powers. Lastly, Section D provides an example of complete mode switching in a random laser.