I recently came across this problem to find the conjugate function of the sum of $r$ smallest elements in a vector
$f(x) = \sum_{i=n-r+1}^{n}x_{[i]}$, $x \in \mathbb{R}^n$
where $x_{[i]}$ is the $i^{th}$ largest element in vector $x$.
My first approach is as follows:
We consider the following function in two cases: $$ y^Tx - f(x) $$ If $y$ has component $y_i \ge 1$ for some $i$, we can construct $x$ such that $x_i$ is very positive ($x_i = t$ where $t >> 0$), and other components of $x$ are all zeros, then $$ y^Tx - f(x) = y_it \rightarrow \infty $$ If $y$ has component $y_i < 1$ for some $i$, we can construct $x$ such that $x_i$ is very negative ($x_i = -t$ where $t >> 0$), and other components of $x$ are all zeros, then $$ y^Tx - f(x) = -y_it + t = (1 - y_i)t \rightarrow \infty $$Therefore, $$ f^*(y) = \infty $$ I wonder if this is the right solution. Any idea is highly appreciated. Thank you all!