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

TL;DR

This paper advances the convex conic reformulation of geometric nonconvex conic optimization problems by analyzing key assumptions and proposing conditions for exact solutions of a broad class of quadratic programs.

Contribution

It provides necessary and sufficient conditions for a crucial assumption in the framework and introduces a new class of quadratic programs solvable via semidefinite relaxation.

Findings

Necessary and sufficient conditions for the key assumption co$(\\mathbb{K} \\cap \\mathbb{J}) = \\mathbb{J}$.

A new class of quadratically constrained quadratic programs solvable exactly by semidefinite relaxation.

Enhanced understanding of convex conic reformulation in nonconvex optimization.

Abstract

A geometric nonconvex conic optimization problem (COP) was recently proposed by Kim, Kojima and Toh as a unified framework for convex conic reformulation of a class of quadratic optimization problems and polynomial optimization problems. The nonconvex COP minimizes a linear function over the intersection of a nonconvex cone $\mathbb{K}$, a convex subcone $\mathbb{J}$ of the convex hull co$\mathbb{K}$ of $\mathbb{K}$, and an affine hyperplane with a normal vector $H$. Under the assumption co$(\mathbb{K} \cap \mathbb{J}) = \mathbb{J}$, the original nonconvex COP in their paper was shown to be equivalently formulated as a convex conic program by replacing the constraint set with the intersection of $\mathbb{J}$ and the affine hyperplane. This paper further studies some remaining issues, not fully investigated there, such as the key assumption co$(\mathbb{K} \cap \mathbb{J}) = \mathbb{J}$ in the framework. More specifically, we provide three sets of necessary-sufficient conditions for the assumption. As an application, we propose a new wide class of quadratically constrained quadratic programs with multiple nonconvex equality and inequality constraints that can be solved exactly by their semidefinite relaxation.