  Let $n$ be a positive integer and $m$ be a positive even integer. Let
${\mathcal A}$ be an $m^{th}$ order $n$-dimensional real weakly symmetric
tensor and ${\mathcal B}$ be a real weakly symmetric positive definite tensor
of the same size. $\lambda \in R$ is called a ${\mathcal B}_r$-eigenvalue of
${\mathcal A}$ if ${\mathcal A} x^{m-1} = \lambda {\mathcal B} x^{m-1}$ for
some $x \in R^n \backslash \{0\}$. In this paper, we introduce two
unconstrained optimization problems and obtain some variational
characterizations for the minimum and maximum ${\mathcal B}_r$--eigenvalues of
${\mathcal A}$. Our results extend Auchmuty's unconstrained variational
principles for eigenvalues of real symmetric matrices. This unconstrained
optimization approach can be used to find a Z-, H-, or D-eigenvalue of an even
order weakly symmetric tensor. We provide some numerical results to illustrate
the effectiveness of this approach for finding a Z-eigenvalue and for
determining the positive semidefiniteness of an even order symmetric tensor.
