A $\mathcal{VU}$-calculus for composite functions and the $\mathcal{U}$-Hessian of partly smooth functions

We assemble a finite-dimensional \(\mathcal{VU}\)-calculus for composite nonsmooth functions whose outer function is convex: the chain rule, separable and convex sums, a strictly differentiable perturbation of a lower semicontinuous (lsc) term, the model \(δ_X+f_0+θ\circ F\), and a finite maximum of