 Copositive reformulation of an orthant Rayleigh minimum

Let $\alpha =\min \{x^{T}Qx:x\ge 0,\Vert x{\Vert }_2=1\}$ for a real symmetric matrix $Q$. Compactness gives attainment. Homogeneity shows $Q-\lambda I$ is a copositive matrix exactly when $\lambda \le \alpha$, provingIts conic program dual is the trace-normalized completely positive optimization problem. A minimizing vector gives $X=x{x}^T$, which certifies equality and dual attainment directly. In general $\alpha$ differs from the unrestricted smallest eigenvalue.