I'm using an alternate optimization scheme to optimize a function $F(a,b,c) = G(a,b,c) + H(c)$ where G is continuous and differentiable, H not differentiable but lower semi-continuous, and proximable. As such, proximal gradient descent is used as a step of the alternate optimization.

I'm interested in the overall convergence of this scheme. In particular, knowing if the set-valued function $A_1 : (a,b) \mapsto c^*$ where $c^* = prox(c^* - \gamma \nabla G(c^*))$ is itself lower semicontinous would provide weak convergence guarantees. But I can't manage to prove this. Is there a specific result I should use, or a paper or textbook I should read?

Sorry if this makes only little sense (especially the title), this is not my usual field of research.