Further Development in Convex Conic Reformulation of Geometric Nonconvex Conic Optimization Problems
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866913494656352256 |
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| author | Arima, Naohiko Kim, Sunyoung Kojima, Masakazu |
| author_facet | Arima, Naohiko Kim, Sunyoung Kojima, Masakazu |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_05922 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Further Development in Convex Conic Reformulation of Geometric Nonconvex Conic Optimization Problems Arima, Naohiko Kim, Sunyoung Kojima, Masakazu Optimization and Control 90C20, 90C22, 90C25, 90C26 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. |
| title | Further Development in Convex Conic Reformulation of Geometric Nonconvex Conic Optimization Problems |
| topic | Optimization and Control 90C20, 90C22, 90C25, 90C26 |
| url | https://arxiv.org/abs/2308.05922 |