  For a Hermitian matrix $H \in \mathbb C^{n,n}$ and symmetric matrices $S_0,
S_1,\ldots,S_k \in \mathbb C^{n,n}$, we consider the problem of computing the
supremum of $\left\{ \frac{v^*Hv}{v^*v}:~v\in \mathbb C^{n}\setminus
\{0\},\,v^TS_iv=0~\text{for}~i=0,\ldots,k\right\}$. For this, we derive an
estimation in the form of minimizing the second largest eigenvalue of a
parameter depending Hermitian matrix, which is exact when the eigenvalue at the
optimal is simple. The results are then applied to compute the eigenvalue
backward errors of higher degree matrix polynomials with T-palindromic,
T-antipalindromic, T-even, T-odd, and skew-symmetric structures. The results
are illustrated by numerical experiments.
