I am reading about how a maximization problem can be converted into a minimization problem: what I have understood is the following:
To change a max problem to a min problem, just multiply the objective function by −1.
1) But I am now confused. I noticed some people have considered that $max_{x}{f(x)}$ is equivalent to $-min_x (-f(x))$. So I am wondering why it is not just $min_x (-f(x))$ ? When we should put a negative sign before the $min$? and when we should not?
2) According to 1), consider now the problem $max_Y ||X - Y||_F^2$ (where $X$, $Y$ are two matrices). So what is its equivalent minimization problem? Is it $min_Y -||X-Y||_F^2$ or $-min_Y -||X-Y||_F^2$ ?
Your help is very appreciated.
It should always have the negative sign in front of it because otherwise it's a minimum, not a maximum. Consider the problem of maximizing $f(x)=-x^2 + 1$ for example which has maximum $1$ but $\min_x(-f(x))=-1$ which is not $1$.