I am trying to find a maximum of quadratic function bounded above/below. The problem is formulated as
\begin{align} &\underset{\mathbf{x}}{\max}~\mathbf{x}^T \mathbf{A} \mathbf{x} \label{eq:16a} \\ &\text{s.t.}\, \left. \begin{array}{l} |\mathbf{x}| \preceq \mathbf{1} \end{array} \right. \label{eq:16b} \end{align} where $\mathbf{A}$ is positive semi-definite.
The Lagrangian of this function is \begin{equation} \mathcal{L} = \mathbf{x}^T \mathbf{A} \mathbf{x} + {\lambda}_-^T (\mathbf{x}+\mathbf{1})-\lambda_+^T(\mathbf{x}-\mathbf{1}), \end{equation} where ${\lambda}_-^T$ and ${\lambda}_+^T$ are Lagrangian multipliers. I am struggling with how to find the optimal $\mathbf{x}^*$ maximizing the objective. Thank you!