Implicit function theorem for subdifferentiable convex functions

Question

I am trying to find a method to apply the implicit function theorem for subdifferential convex functions. The original theorem provides an equation for the partial derivative of the implicit function w.r.t $x$ by using the chain rule. There are some variants for non-differentiable functions but these only guarantee that a solution $x=x(y)$ to the equation $$ F\big(x(y),y\big) = 0 $$ exists and do not give any information about the partial derivative of $x$ respect to $y$. Is there any way to adapt this in terms of subdifferentials instead of partial derivatives?

Thanks

Answer 1

I do not know if your question has an answer, if it has please share it !

With the regular subderivative you might have trouble. But the notion of "conservative field" is a generalisation of the derivative that shows some nice properties. see https://link.springer.com/article/10.1007/s10107-020-01501-5 and https://proceedings.neurips.cc/paper_files/paper/2021/file/70afbf2259b4449d8ae1429e054df1b1-Paper.pdf . For instance the chain rule is rigorously obtained for non differentiable functions that have a conservative field.

In short a concervative field is a set value mapping that is equal almost everywhere to the differential of your function and that verifies some other properties such as closedness of its graph. Only functions that are differentiable almost everywhere can have conservative fields, that is the case for all convex functions. The subdifferential of a convex function is a conservative field for the same function.

This might help you but I do not garantee that it will be sufficient.

Answer 2

I realised there is an answer in "Optimization and nonsmooth analysis" Frank H. Clarke page 255 . The answer uses the clarke subdifferential (it corresponds to the subdifferential of a convex function).