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Just beginning to learn convex analysis and optimization, I have some inquiries to make with regard to the absolute value function $f(x)= |x|$. This function is clearly convex, but since we know that $|x|= \max\{x,-x\}$, does this make $\max$ a convex function? How about $\min$? My guess is that the minimum function is concave, but I might be wrong. I'd appreciate any helpful input on these questions I'm having.

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2 Answers 2

up vote 2 down vote accepted

The optimisation result is that the pointwise supremum of affine functions is convex. On the contrary, the pointwise infimum of linear (or affine) functions is concave.

$\max\{x,-x\}$ is convex because it is the maximum of two linear functions of $x$, namely $x$ and $-x$.

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You are right about the maximum, and you could find this fact in Wikipedia: properties of convex functions.

Also right about the minimum being concave. Remember that everything we say about convex functions translates into statements about concave functions simply by replacing $f$ with $-f$. This also switches $\max$ and $\min$ around.

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So $\min$ is never convex? –  Libertron Feb 19 '13 at 0:13
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@Sachin Depends on $\min$ of what you take. For example, $\min(e^x,1+x)$ is convex. –  user53153 Feb 19 '13 at 0:15

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