= Solution
For $X$ in the <dual cone> $C$, copositivity of $Q-\lambda I$ gives $\langle X,Q-\lambda I\rangle_F\geq0$. An upper-bounding <Lagrangian> for the maximization is therefore
$$
L(\lambda,X)=\lambda+\langle X,Q-\lambda I\rangle_F
=\langle Q,X\rangle_F+\lambda(1-\operatorname{tr}X).
$$
Its <supremum> over unrestricted $\lambda$ is finite precisely when $\operatorname{tr}X=1$. Minimizing that upper bound gives the <conic program>
$$
\boxed{\min_{X\in C}\ \langle Q,X\rangle_F
\quad\text{subject to }\operatorname{tr}X=1}.
$$
This is <completely positive optimization>, not simply <semidefinite programming>: $C$ is the <completely positive cone>.
There is also a direct equality certificate. Every feasible $X=\sum_j u_ju_j^T$, $u_j\geq0$, has $\sum_j\|u_j\|_2^2=1$. Its objective is a weighted average of the nonnegative-sphere <Rayleigh quotients>, hence is at least $\alpha$. For a minimizing unit <vector> $x_*$ from part (g), $X_*=x_*x_*^T$ is feasible and has objective $\alpha$. Thus $\boxed{X_*=x_*x_*^T}$ attains the dual and both values agree, without relying on unverified regularity assumptions.
Back to article page
