I have the following optimization problem,
$ \arg\min_\mathbf{a} f(\max(\mathbf{a}^T\mathbf{b}_i))\;\; i=1, \dots ,N$
where $\mathbf{a}$ and $\mathbf{b}_i$ are vectors of dimension $d$. Let $B = \left[\mathbf{b}_1, \dots, \mathbf{b}_N\right]$.
Can I write the above optimization problem as
$\arg\min f(\mathbf{a}^T \mathbf{c})$
where $\mathbf{c}$ contains the elements which are the rowwise maximum of $B$.
e.g. $B = \begin{bmatrix} 1 & 2 & 4 \\ 5 & 1 & 2 \\ 1 & 4 & 1 \end{bmatrix} \implies \mathbf{c} = \begin{bmatrix} 4 \\ 5 \\ 4 \end{bmatrix}$
If $d = 1$, and $a$ is positive, assume $\mathbf{b} = [2,-1, 4, 5]$
$\arg\min f(\max(a\left[2,-1, 4, 5\right])) = \arg\min f(a \max \left[2,-1, 4, 5\right])$
But I dont know how can I prove this for vectors ($d>1$) under some assumptions (e.g. all the elements in $\mathbf{a}$ or $B$ are positive?