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A geometric nonconvex conic optimization problem (COP) was recently proposedby Kim, Kojima and Toh as a unified framework for convex conic reformulation ofa class of quadratic optimization problems and polynomial optimizationproblems. The nonconvex COP minimizes a linear function over the intersectionof a nonconvex cone $\mathbb{K}$, a convex subcone $\mathbb{J}$ of the convexhull co$\mathbb{K}$ of $\mathbb{K}$, and an affine hyperplane with a normalvector $H$. Under the assumption co$(\mathbb{K} \cap \mathbb{J}) = \mathbb{J}$,the original nonconvex COP in their paper was shown to be equivalentlyformulated as a convex conic program by replacing the constraint set with theintersection of $\mathbb{J}$ and the affine hyperplane. This paper furtherstudies some remaining issues, not fully investigated there, such as the keyassumption co$(\mathbb{K} \cap \mathbb{J}) = \mathbb{J}$ in the framework. Morespecifically, we provide three sets of necessary-sufficient conditions for theassumption. As an application, we propose a new wide class of quadraticallyconstrained quadratic programs with multiple nonconvex equality and inequalityconstraints that can be solved exactly by their semidefinite relaxation.

Further Development in Convex Conic Reformulation of Geometric Nonconvex Conic Optimization Problems