What is the dual function of the $f(x) = \sum_i x_i \ln(x_i)$?

Question

I have always assumed that the dual of $$f(x) = \sum_i x_i \ln(x_i), x \in \Delta^N$$ is $$f(y) = \ln(\sum_i e^{y_i}), y \in \mathbb{R}^N$$ (and vice versa, the other direcion is extremely well known)

However, in MM Optimization Algorithms by Kenneth Lange, it writes:

However, notice the domain is $x \in \mathbb{R}^m$

Can anyone please confirm the dual of neg entropy function $$f(x) = \sum_i x_i \ln(x_i), x \in \Delta^N$$

References are much appreciated

Answer 1

You are both correct. There are two conjugate pairs: $$ \begin{align*} f(x) &= \begin{cases} \sum_i x_i \ln(x_i) & \text{if } x \in \Delta^N \\ \infty & \text{else}\end{cases} \\ f^*(y) &= \log\left(\sum_i e^{y_i}\right) \end{align*} $$ and $$ \begin{align*} g(x) &= \begin{cases} \sum_i x_i \ln(x_i) & \text{if } x \in \mathbb{R}^n_+ \\ \infty & \text{else}\end{cases} \\ g^*(y) &= \sum_i e^{y_i-1} \end{align*} $$ See, e.g., Table 1.