# Clarification about the relation between maximization and minimization of objective functions

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 $$-min_Y -||X-Y||_F^2$$ ?

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 $$1$$.