I have a function $f: \mathbb{R} \to \mathbb{R}_+$ defined by $$ f(x) = f_1(x) + f_2(x) + f_3(x) - f_4(x) $$ where every $f_i$ is a proper, closed convex function defined over some interval $[a,b] \subset \mathbb{R}$. I would like to minimize $f$
$$ \arg\min\limits_{x \in [a, b]} f(x) $$
The sum of a set of convex functions is convex, so I thought of approaching this by defining $$ g(x) = f_1(x) + f_2(x) + f_3(x) $$ and $h(x) = f_4(x)$ and minimizing via the difference of convex functions algorithm (DCA) (see slide 34).
$$ \arg\min\limits_{x \in [a, b]} g(x) - h(x) $$
This requires the evaluation of the convex conjugate $g^\star$ of $g$:
$$ g^\star(y) = \mathrm{sup}_{x} \{ \langle x, y \rangle - f(x) \}$$
I know the conjugate functions $f_i^\star$ of each of my constitutive functions $f_i$. However, my understanding is that in general:
$$ (f_1 + f_2 + f_3)^\star (y) \neq (f_1^\star + f_2^\star + f_3^\star)(y)$$
Apparently, extrapolating from this question (and reference they give):
$$ (f_1 + f_2 + f_3)^\star(y) = \inf\limits_{x = x_1 + x_2 + x_3} \{ f_1^\star(x_1) + f_2^\star(x_2) + f_3^\star(x_3)\}$$
Alternatively, extracting from this answer, the solution can be obtained via infimal convolution:
$$ (f_1 \Box f_2 \Box f_3)(y) = \sup_{x} \sup_{x = x_1+x_2+x_3} \{ \langle x, y \rangle - f_1(x_1) - f_2(x_2) - f_3(x_3)\}$$
Incidentally, I know the Moreau envelopes of each of the $f_i's$ individually... but I don't know how to resolve the inner $\mathrm{sup}$/obtain $x_1, x_2, x_3$, given some value $x$.... what am I missing?