# 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 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 $f_2^*$?

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To get the correct spacing, use \langle and \rangle, which produce, respectively, $\langle$ and $\rangle$, instead of < and >. –  Arturo Magidin Nov 12 '11 at 23:32