Coercive function

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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

\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):

|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
y^TAy \ge a||y||^2.

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