= Copositive reformulation of an orthant Rayleigh minimum
Let $\alpha=\min\{x^TQx:x\geq0,\ \|x\|_2=1\}$ for a real <symmetric matrix> $Q$. Compactness gives attainment. Homogeneity shows $Q-\lambda I$ is a <copositive matrix> exactly when $\lambda\leq\alpha$, proving
$$
\alpha=\max\{\lambda:Q-\lambda I\in\operatorname{COP}_n\}.
$$
Its <conic program> dual is the trace-normalized <completely positive optimization> problem. A minimizing <vector> gives $X=xx^T$, which certifies equality and dual attainment directly. In general $\alpha$ differs from the unrestricted smallest <eigenvalue>.
Back to article page
