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The convex cone of completely positive matrices:It is a closed convex cone and the dual cone of the copositive cone under the Frobenius inner product. The finite-sum definition imposes no closure by fiat; closedness of the completely positive cone supplies that fact.

Optimization of a linear function subject to affine constraints and membership in a closed convex cone. Choosing the positive semidefinite cone gives semidefinite programming; other choices include copositive optimization and completely positive optimization. Dual cones produce scalar-product bounds on feasible objectives.

For a closed convex cone $C$ in a finite-dimensional real inner product space and $z\notin C$, there exists $y$ with $⟨y,z⟩<0$ and $⟨y,c⟩\ge 0$ for all $c\in C$. Here is a direct proof. A nearest point $c_0\in C$ exists by compactness after restricting to a sufficiently large ball. Differentiating squared distance along the segment towards $c\in C$ gives $⟨c_0-z,c-c_0⟩\ge 0$. Using $c=0$ and $c=2c_0$ shows $⟨c_0-z,c_0⟩=0$. Set $y=c_0-z\ne 0$: then $⟨y,c⟩\ge 0$ and $⟨y,z⟩=-\Vert y{\Vert }^2<0$. This is a conic form of the Hahn-Banach separation theorem.