Let $g:\mathbb{R}^n\to\mathbb{R}^m$ be linear defined by $g(x) = Ax$. I'm looking for an analog of the adjoint relationship: $\langle\lambda,g(x)\rangle=\langle\lambda,Ax\rangle=\langle A^\top\lambda,x\rangle$ for the case where $g(x)$ is not linear.
For example, suppose we have that $h:\mathbb{R}^n\to\mathbb{R}^m$ with components $h_i(x)=\max\bigl\{\langle a_i,x\rangle, \langle b_i,x\rangle\bigr\}$ is piecewise linear convex, for $i=1,\ldots,m$. Then we should have the "adjoint" relationship $$ \langle\lambda,h(x)\rangle = \langle C(x)^\top\!\lambda,x\rangle $$ where the $i$th row of $C(x)$ is given by $$ c_i(x)=\begin{cases}a_i,\quad \text{if $a_i^\top x>b_i^\top x$}\\b_i,\quad\text{o.w.}\end{cases} $$
Are there other relationships like this? Do relationships like this have a name? Finally, can we interpret $\max_x \sum_{i=1}^m \lambda_i h_i(x)-f(x)$ as the conjugate \begin{align} \max_x\bigl\{\langle C(x)^\top\lambda,x\rangle-f(x)\bigr\}=f^*(C(x^*)^\top\lambda), \end{align} where $x^*$ is the maximizer?