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Use the conic Carathéodory theorem in the real space of symmetric matrices, of dimension $N=n(n+1)/2$. A convergent sequence $B_{\ell }$ in the completely positive cone has padded factorizations $B_{\ell }={\sum }_{j=1}^N{x}_{\ell j}{x}_{\ell j}^T$ with nonnegative factors. Their total squared norms equal $\mathrm{tr}{B}_{\ell }$ and are uniformly bounded. A simultaneous convergent subsequence of the finite factor tuple gives $B={\sum }_j{x}_j{x}_j^T$ with $x_j\ge 0$. Thus the limit remains in the cone. The argument also bounds the required number of factors by $N$.

A real symmetric matrix admitting a finite sum of nonnegative rank-one matrices $x_j{x}_j^T$. It is both a positive semidefinite matrix and a nonnegative matrix, but these two properties alone need not imply membership in the completely positive cone in arbitrary dimension. This matrix notion is distinct from a completely positive map.

Conic optimization using the completely positive cone. For a real symmetric matrix $Q$, the trace-normalized program $\min \{⟨Q,X{⟩}_F:X\in {\mathrm{CP}}_n,\mathrm{tr}X=1\}$ minimizes a weighted average of nonnegative-unit-vector Rayleigh quotients. The weights are the squared norms of the factors in $X={\sum }_j{x}_j{x}_j^T$. Hence a minimizing rank-one factor attains the same value as the original orthant minimum.

For $X$ in the dual cone $C$, copositivity of $Q-\lambda I$ gives $⟨X,Q-\lambda I{⟩}_F\ge 0$. An upper-bounding Lagrangian for the maximization is thereforeIts supremum over unrestricted $\lambda$ is finite precisely when $\mathrm{tr}X=1$. Minimizing that upper bound gives the conic programThis is completely positive optimization, not simply semidefinite programming: $C$ is the completely positive cone.

There is also a direct equality certificate. Every feasible $X={\sum }_j{u}_j{u}_j^T$, $u_j\ge 0$, has ${\sum }_j\Vert u_j{\Vert }_2^2=1$. Its objective is a weighted average of the nonnegative-sphere Rayleigh quotients, hence is at least $\alpha$. For a minimizing unit vector $x_{\ast }$ from part (g), $X_{\ast }=x_{\ast }{x}_{\ast }^T$ is feasible and has objective $\alpha$. Thus $\boxed{X_{\ast }=x_{\ast }{x}_{\ast }^T}$ attains the dual and both values agree, without relying on unverified regularity assumptions.