Characterizing nonconvex boundaries via scalarization
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917004755075072 |
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| author | Ma, Jin Xia, Weixuan Zhang, Jianfeng |
| author_facet | Ma, Jin Xia, Weixuan Zhang, Jianfeng |
| contents | We present a unified approach for characterizing the boundary of a possibly nonconvex domain. Motivated by the well-known Pascoletti--Serafini method of scalarization, we recast the boundary characterization as a multi-criteria optimization problem with respect to a local partial order induced by a spherical cone with varying orient. Such an approach enables us to trace the whole boundary and can be considered a general dual representation for arbitrary (nonconvex) sets satisfying an exterior cone condition. We prove the equivalence between the geometrical boundary and the scalarization-implied boundary, particularly in the case of Euclidean spaces and two infinite-dimensional spaces for practical interest. By reformulating each scalarized problem as a parameterized constrained optimization problem, we shall develop a corresponding numerical scheme for the proposed approach. Some related applications are also discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_09918 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterizing nonconvex boundaries via scalarization Ma, Jin Xia, Weixuan Zhang, Jianfeng Optimization and Control Metric Geometry 90C26, 90C29, 93E20 We present a unified approach for characterizing the boundary of a possibly nonconvex domain. Motivated by the well-known Pascoletti--Serafini method of scalarization, we recast the boundary characterization as a multi-criteria optimization problem with respect to a local partial order induced by a spherical cone with varying orient. Such an approach enables us to trace the whole boundary and can be considered a general dual representation for arbitrary (nonconvex) sets satisfying an exterior cone condition. We prove the equivalence between the geometrical boundary and the scalarization-implied boundary, particularly in the case of Euclidean spaces and two infinite-dimensional spaces for practical interest. By reformulating each scalarized problem as a parameterized constrained optimization problem, we shall develop a corresponding numerical scheme for the proposed approach. Some related applications are also discussed. |
| title | Characterizing nonconvex boundaries via scalarization |
| topic | Optimization and Control Metric Geometry 90C26, 90C29, 93E20 |
| url | https://arxiv.org/abs/2510.09918 |