I am doing the additional exercises of the book Convex Optimization, and I am stuck at question 3.32 (page 29 of this pdf).
The problem is:
$$ \begin{aligned} & \text{minimize} && f_0(x)\\ & \text{subject to} && \sum_i^m(1 + \lambda f_i(x))_+ \le m - k\\ &&& \lambda > 0 \end{aligned} $$
where $(\cdot )_+$ represents $\max\{\cdot,\ 0\}$, and $f_0, f_i$ are convex functions.
The author gives a hint that, to solve this problem using convex optimization, we may use the change of variables.
My question is, as far as I know, this problem is already a convex optimization problem, because the expression
$$ \sum_i^m(1 + \lambda f_i(x))_+ - m + k $$
is convex for both $\lambda$ and $x$. Then what's the point of changing variables here?