# Coercive function

The function $LaTeX: f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is coercive with respect to $LaTeX: X \subseteq \mathbb{R}^n$ if there exists a vector $LaTeX: x^0 \in X$ such that for any sequence $LaTeX: x^k \in X$ having the property that $LaTeX: \|x^k \| \rightarrow \infty$, we also have

$LaTeX: \lim_{k \rightarrow \infty} | (f(x^k) - f(x^0))^T (x^k - x^0) | = \infty,$
where any norm can be used. This arises in variational inequalities (and complementarity problems).

Some people use a different definition, where $LaTeX: f : \mathbb{R}^n \rightarrow \mathbb{R}$ (i.e., a scalar, real-valued function):

$LaTeX: |f(x)| / \| x \| \rightarrow \infty, \;\mbox{ as }\; \|x\| \rightarrow \infty$
Note that the two definitions differ, even for $LaTeX: n=1$. For example, $LaTeX: f(x)=|x|$ is coercive under the first definition and is not under the second. For the bilinear function, $LaTeX: f(x,y)=x^TAy$, for some square matrix $LaTeX: A$, $LaTeX: f$ is coercive if there exists $LaTeX: a > 0$ such that
$LaTeX: y^TAy \ge a||y||^2.$