It is shown that the well-known problem of determining the probability of extinction in a simple branching process has a duality relation to the problem of determining that offspring distribution which is in a sense closest to the original one and for which the new process is subcritical (or critical). The latter problem is also considered with respect to various measures of distance.
"The Geometric Programming Dual to the Extinction Probability Problem in Simple Branching Processes." Ann. Probab. 9 (3) 498 - 503, June, 1981. https://doi.org/10.1214/aop/1176994422