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Hi guys just wanted to clear up this notion of local/global extremum at an interior point. I would like to know what's the proper definition for a function having a local/global extremum.

I know a local extremum can be a maximum or minimum value while a global extremum can be the largest maximum or minimum value. As it relates , to defining it is there more to it or is this the general idea.

$f$ has a global maximum at $x$ if $f(y) \leq f(x)$, where $f(y$) is the function values.

The opposite is for the minimum where $f(x) \leq f(y)$ and for the minimum a neighborhood is considered in each instance within the function.

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John
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  • Generally a problem has a feasible set $F$ (it may be the whole space). If $f$ has a global $\max$ at $x$ then $f(x) \ge f(y)$ for all $y \in F$. If $f$ has a local $\max$ at $x$ then there is some neighbourhood $B$ of $x$ such that $f(x) \ge f(y)$ for all $y \in F \cap B$. Same for $\min$. Clearly a global $\max$ is a local $\max$. – copper.hat Oct 07 '19 at 17:06
  • so saying the function has a local or global extremum is really just saying it has a local or global maximum or minimum at an interior point of the set ? @copper.hat – John Oct 07 '19 at 17:10
  • Not in general. There is no mention of interior of set. For example, if the problem is $P:\ \ \min {(x-1)(x-2)(x-3) | x \ge 0 }$ then the global $\min$ is at $x=0$, which is not in the interior. – copper.hat Oct 07 '19 at 17:20
  • The feasible set need not even have an interior. – copper.hat Oct 07 '19 at 17:21

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