Suppose that $A\in \mathbb R^{n\times n}$ is a symmetric matrix with exactly one negative eigenvalue which has corresponding eigenvector $v$. Consider the problem of minimizing $x^TAx$ over the set $\{x\in\mathbb R^{n\times n}:x\geq 0\text{ entry-wise and }v^Tx = 1\}$ (assuming that $v$ has a positive component).
How should I go about solving this? Does anyone know an approachable textbook about this type of subject?
With an eigenvalue decomposition $A = VDV^T$ and introducing $z = V^Tx$ you are minimizing $\sum_k d_{k} z_k^2$ over $Vz\geq 0, v_i^TVz = 1$ where $i$ is the index of the negative eigenvalue. The equality simply says $z_i = 1$ and thus you are solving $\sum_{k\neq i} d_{k} z_k^2 + d_i$ over $Vz\geq 0, v_i^TVz = 1$, whch is a standard convex QP.