Copositive optimization

Conic optimization using the copositive cone. The constraint $Q-\lambda I\in {\mathrm{COP}}_n$ supplies an exact reformulation of a Rayleigh quotient minimum on the nonnegative orthant. Exact conic formulation does not itself provide an efficient membership algorithm.

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.