In an optimization problem that depends on parameters, an important issue is the effect that perturbations of the parameters can have on solutions to the problem and their associated multipliers. Under quite broad conditions the possibly multi-valued mapping that gives these elements in terms of the parameters turns out to enjoy a property of “proto-differentiability.” Generalized derivatives can then be calculated by solving an auxiliary optimization problem with auxiliary parameters. This is constructed from the original problem by taking second-order epi-derivatives of an essential objective function.