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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.10421 |
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| _version_ | 1866918200791269376 |
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| author | Kabgani, Alireza Ahookhosh, Masoud |
| author_facet | Kabgani, Alireza Ahookhosh, Masoud |
| contents | This paper studies the fundamental properties of the high-order forward-backward splitting mapping (HiFBS) and its associated forward-backward envelope (HiFBE) through the lens of high-order regularization for nonconvex composite functions. Specifically, we (i) establish the boundedness and uniform boundedness of HiFBS, along with the Hölder and Lipschitz continuity of HiFBE; (ii) derive an explicit form for the subdifferentials of HiFBE; and (iii) investigate necessary and sufficient conditions for the differentiability and weak smoothness of HiFBE under suitable assumptions. By leveraging the prox-regularity of $g$ and the concept of $p$-calmness, we further demonstrate the local single-valuedness and continuity of HiFBS, which in turn guarantee the differentiability of HiFBE in neighborhoods of calm points. This paves the way for the development of gradient-based algorithms tailored to nonconvex composite optimization problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_10421 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On fundamental properties of high-order forward-backward envelope Kabgani, Alireza Ahookhosh, Masoud Optimization and Control This paper studies the fundamental properties of the high-order forward-backward splitting mapping (HiFBS) and its associated forward-backward envelope (HiFBE) through the lens of high-order regularization for nonconvex composite functions. Specifically, we (i) establish the boundedness and uniform boundedness of HiFBS, along with the Hölder and Lipschitz continuity of HiFBE; (ii) derive an explicit form for the subdifferentials of HiFBE; and (iii) investigate necessary and sufficient conditions for the differentiability and weak smoothness of HiFBE under suitable assumptions. By leveraging the prox-regularity of $g$ and the concept of $p$-calmness, we further demonstrate the local single-valuedness and continuity of HiFBS, which in turn guarantee the differentiability of HiFBE in neighborhoods of calm points. This paves the way for the development of gradient-based algorithms tailored to nonconvex composite optimization problems. |
| title | On fundamental properties of high-order forward-backward envelope |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2511.10421 |