# Rewriting a maximization problem to a minimization one

Suppose that the function $$g: \mathbb{R}^n\rightarrow\mathbb{R}$$ is convex on $$\mathbb{R}^n$$ and that $$\mathbf{d}\in\mathbb{R}^n$$. Is the problem to
$$maximize -\sum^n_{j=1}x^2_j$$
subject to conditions (shouldn't be relevant)
a convex problem?

I have determined that the conditions are convex but have a hard time digesting the goal function.

In the solution the following is stated about the objective function:

It is clear that the objective function can be written as the minimization of a (strictly) convexfunction.

I realise that convexity is only relevant to a minimization problem (atleast in my case) but I dont see how the problem could be rewritten to be a minimization problem of a convex function.

When solving the problem the first time around I tried thinking about the problem in $$\mathbb{R}^3$$ and $$\mathbb{R}^2$$ and could only come to the conclusion that as a maximization problem it is strictly concave.

How should i go about thinking when it comes to rewriting a maximization problem to a minimization one?

Given some objective function $$f : \mathbb R^n \to \mathbb R$$, the optimization problem $$\max -f(x)$$ can be re-written as $$\min f(x)$$ and vice versa. Note that all minima of $$f(x)$$ are maxima of $$-f(x)$$. Also, if $$f$$ is concave, then $$-f$$ is convex, and vice versa.
The objective function $$f(x) = -\sum^n_{j=1}x^2_j$$ is concave in its domain, and so $$\sum^n_{j=1}x^2_j = -f(x)$$ is convex.