In mathematics, a radially unbounded function is a function
f :
n
for which [1]
Or equivalently,
Such functions are applied in control theory and required in optimization for determination of compact spaces.
Notice that the norm used in the definition can be any norm defined on
, and that the behavior of the function along the axes does not necessarily reveal that it is radially unbounded or not; i.e. to be radially unbounded the condition must be verified along any path that results in:
For example, the functions
{\displaystyle {\begin{aligned}f_{1}(x)&=(x_{1}-x_{2})^{2}\\f_{2}(x)&=(x_{1}^{2}+x_{2}^{2})/(1+x_{1}^{2}+x_{2}^{2})+(x_{1}-x_{2})^{2}\end{aligned}}}
are not radially unbounded since along the line
, the condition is not verified even though the second function is globally positive definite.
This mathematical analysis–related article is a stub.
You can help Wikipedia by expanding it.