#convex-conjugate

Question 1: Single choice

The convex conjugate of $f$ is $f^*(y) = \sup_x \big(y^\top x - f(x)\big)$ . What is the conjugate of the log-sum-exp function $f(x) = \log\sum_{j=1}^{d} e^{x_j}$ on $\mathbb{R}^d$ ?

$f^*(y) = \log\sum_j e^{y_j}$ for every $y$ ; log-sum-exp is its own conjugate1
$f^*(y) = -\sum_j y_j \log y_j$ for every $y$ with positive entries2
$f^*(y) = \sum_j y_j \log y_j$ if $y$ is a probability vector, and $+\infty$ otherwise3
$f^*(y) = \max_j y_j$ for every $y$ 4

Timed test