TRUST REGION SUBPROBLEMS
\[ B = B^t ~~{\rm real}~ (n-1) \times (n-1)   \]
eigenvalues: $\gamma_1 \geq \cdots \geq \gamma_{n-1}$
\begin{eqnarray*}
 v(x) = x^t B x   \\
 {\rm subject ~ to}~  x^tx -1 =0
\end{eqnarray*}

Lagrangian: $L(x,\lambda) = v(x) - \lambda(x^tx -1)$

Lagrange multipliers, stationary points $\iff$ eigenvalues, eigenvectors\\
~~~~~~~~~~$Bx - \lambda x = 0 $~~~~~~~~~~~~~~~~ $Bu_i = \lambda_iu_i$

For example
\[ \lambda_1 = max_{x^tx=1} v(x) = v(u_1)  \]
\[ \lambda_{n-1} = min_{x^tx=1} v(x) = v(u_{n-1})  \]

(no convexity required)