I am attempting to solve the argument maximization problem


$$\arg\sup\_x \{ \langle x,l \rangle − f\_1(x)−f\_2(x) \} \ \ \ \ \ \ \ \ \ \ (1)$$


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. We can further assume that $f^∗\_1$ and $f^∗\_2$ are differentiable and that we can evaluate their gradients. 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


$$\sup\_x\{⟨x,l⟩−f\_1(x)−f\_2(x)\}=\inf\_w \ \{ f^\*\_1(l−w) + f^∗\_2(w) \}.$$


Is it possible to compute the solution to problem $(1)$ is an analogous manner, making only calls to the conjugate functions $f^∗\_1$ and $f^∗\_2$ or their gradients?



Is it possible to solve the argument maximization problem $\arg\max_x \langle x,l \rangle −f_1(x)−f_2(x)$ via convex duality?
I ultimately did find a solution to this problem. First, you compute the solution $\bar{w}$ to 


$$\bar{w} \in \arg\inf\_w f^\*\_1(\ell - w) + f\_2^\*(w).$$


A solution $\bar{x}$ to (1) can then be computed by choosing 


$$\bar{x} \in \partial f\_1^\*(\ell-\bar{w}) \cap \partial f\_2^\*(\bar{w})$$


where $\partial f$ denotes the subgradient of $f$. A proof of this fact appears as part of Theorem 15 in this JMLR paper: 


<http://jmlr.org/papers/volume8/rifkin07a/rifkin07a.pdf>


