I am trying to solve the following problem.
Given a $n$-dimension column vector $\mathbf z$ with each element $z_i\in\mathbb R$, I wish to minimize the following objective function.
$\mathbf z^T\cdot f(\mathbf z)-max(z_i)$, where $f$ is the softmax function, $=e^{z_i}/\sum{e^{z_i}}$.
I can only work out the $2$-dimension case $\mathbf z=[z_1, z_2]^T$ by computing the partial derivative of $z_1$ and $z_2$ and set them to $0$.
I want to know if this problem can be converted to some well-studied optimiation problems like LP or GP? Or how can I solve it?