Most algorithms in order restricted statistical inference express the estimates in terms of certain summary statistics computed from pooled samples. These algorithms may or may not yield optimal estimates depending on whether or not the Cauchy mean value property holds strictly for the summary statistics. In this paper a minimum lower sets algorithm, which holds generally, is described and used to prove the optimality of estimates described by a max-min formula.
"Algorithms in Order Restricted Statistical Inference and the Cauchy Mean Value Property." Ann. Statist. 8 (3) 645 - 651, May, 1980. https://doi.org/10.1214/aos/1176345014