I am attemping to solve the argument maximization problem
$\arg\sup_x \{\langle x,l\rangle - f_1(x) - f_2(x)\}\qquad\qquad\qquad\qquad (1)$
where the functions $f_1$ and $f_2$ are concave but difficult to evaluate but their convex conjugates $f_1^*$ and $f_2^*$ are easy to evaluate. Since the sum operation is dual to the infimal convolution (or epi-sum) operation $(g\#h)(x) = \inf_w \{g(x-w)+h(w)\},$ the standard maximization problem is easy to compute by duality using the identity $\sup_x \{\langle x,l\rangle - f_1(x) - f_2(x)\} = \inf_w \{f_1^*(l-w)+f_2^*(w)\}.$
Is it possible to compute the solution to problem (1) is an analogous manner, making only calls to the conjugate functions $f_1^*$ and $f_2^*$?