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extremum

Extrema are minima and maxima. The singular forms of these words are extremum, minimum, and maximum.

Extrema may be “global” or “local”. A global minimum of a function $f$ is the lowest value that $f$ ever achieves. If you imagine the function as a surface, then a global minimum is the lowest point on that surface. Formally, it is said that $f\colon U\to V$ has a global minimum at $x$ if $\forall u\in U,f(x)\leq f(u)$ .

A local minimum of a function $f$ is a point $x$ which has less value than all points “next to” it. If you imagine the function as a surface, then a local minimum is the bottom of a “valley” or “bowl” in the surface somewhere. Formally, it is said that $f\colon U\to V$ has a local minimum at $x$ if $\exists$ a neighborhood $N$ of $x$ such that $\forall y\in N$ , $f(x)\leq f(y)$ .

If you flip the $\leq$ signs above to $\geq$ , you get the definitions of global and local maxima.

A “strict local minima” or “strict local maxima” means that nearby points are strictly less than or strictly greater than the critical point, rather than $\leq$ or $\geq$ . For instance, a strict local minima at $x$ has a neighborhood $N$ such that $\forall y\in N$ , $(f(x)<f(y)\textrm{ or }y=x)$ .

A saddle point is a critical point which is not a local extremum.

A related concept is plateau.

Finding minima or maxima is an important task which is part of the field of optimization. This task is also important in Physics where the minima correspond to equilibria.

Defines:

global minima, global maxima, local minima, local maxima, global minimum, global maximum, local minimum, local maximum, strict local minima, strict local maxima, strict local minimum, strict local maximum, saddle point

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extrema

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Definition

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Reference

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