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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?

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1 Answer 1

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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.

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