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

A Geometrical Analysis of a Class of Nonconvex Conic Programs for Convex Conic Reformulations of Quadratic and Polynomial Optimization Problems