  We present a geometrical analysis on the completely positive programming
reformulation of quadratic optimization problems and its extension to
polynomial optimization problems with a class of geometrically defined
nonconvex conic programs and their covexification. The class of nonconvex conic
programs is described with a linear objective functionin a linear space $V$,
and the constraint set is represented geometrically as the intersection of a
nonconvex cone $K \subset V$, a face $J$ of the convex hull of $K$ and a
parallel translation $L$ of a supporting hyperplane of the nonconvex cone $K$.
We show that under a moderate assumption, the original nonconvex conic program
can equivalently be reformulated as a convex conic program by replacing the
constraint set with the intersection of $J$ and the hyperplane $L$. The
replacement procedure is applied to derive the completely positive programming
reformulation of quadratic optimization problems and its extension to
polynomial optimization problems.
