Given $f\left( x\right) =\sum ^{n}_{i=1}x_{i}\log x_{i},x\in \mathbb{R}$. How can I determine the Fenchel conjugate function $f^{\ast }$?
The Fenchel conjugate of f is defined by: $f^{\ast }\left( y\right) =\sup \left\{ \langle y,x\rangle -f\left( x\right) | x\in \overline{\mathbb{R} }\right\} ,y\in \overline{\mathbb{R} }$
I don't see how I can find the supremum of this function.
Let $g_a(t)=at-t\log t$ for $a\in\overline{\mathbb R}, t\in[0,\infty]$, then $g_a'(t)=a-\log t-1$ and $g_a''(t)=-1/t$.
It is easy to show $g_a$ has a unique global maximum at $t=e^{a-1}$, with maximum value $\sup(g_a)=e^{a-1}$.
Now we observe that
$$\sup_x\Big\{\langle y,x\rangle-f(x)\Big\}=\sup_x\left\{\sum_{i=1}^ng_{y_i}(x_i)\right\}=\sum_{i=1}^n\sup_{x_i}\Big\{g_{y_i}(x_i)\Big\}=\sum_{i=1}^ne^{y_i-1}.$$