The ALSIE (Automatic Lens Design By Solving Inequalities, T. Suzuki, et al., J. Opt. Soc. Am., 56, 677, 1966) algorithm is a powerful lens design procedure for the optimization of conventional and unconventional lens systems. However, a problem of the ALSIE algorithm is the selection of an adequate set of boundaries to the performance functions. In this work a procedure of selecting the boundaries is discussed. It is experimentally found, by designing a doublet lens to be used in a multiple imaging lens system, that the selected boundaries dealt to a faster convergence than with the original selection of boundaries.