Given an objective function that is invariant under an action of a Lie group, we study how its subgradients relate to the orbits of the action. Our main finding is that they satisfy projection formulae analogous to those stemming from the Whitney and Verdier stratifications. If the function is definable in an o-minimal structure on the real field, then we also obtain an invariant variational stratification. On the application side, we derive a conservation law for subgradient dynamics under minimal assumptions. It can be used to detect instability in discrete subgradient dynamics.
Cédric Josz (Sun,) studied this question.