Level proximal subdifferential, variational convexity, and pointwise quadratic approximation
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866914493401923584 |
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| author | Luo, Honglin Wang, Xianfu Wang, Ziyuan Yang, Xinmin |
| author_facet | Luo, Honglin Wang, Xianfu Wang, Ziyuan Yang, Xinmin |
| contents | Level proximal subdifferential was introduced by Rockafellar recently for studying proximal mappings of possibly nonconvex functions. In this paper a systematic study of level proximal subdifferential is given. We characterize variational convexity of a function by local firm nonexpansiveness of proximal mappings or local relative monotonicity of level proximal subdifferential, and use them to study local convergence of proximal gradient method and others for variationally convex functions. Variational sufficiency guarantees that proximal gradient method converges to local minimizers rather than just critical points. We also investigate the existence, single-valuedness and integration of level proximal subdifferential, and quantify pointwise quadratic approximation (or Lipschitz smoothness) of a function. As a powerful tool, level proximal subdifferential provides deep insights into variational analysis and optimization. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_00648 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Level proximal subdifferential, variational convexity, and pointwise quadratic approximation Luo, Honglin Wang, Xianfu Wang, Ziyuan Yang, Xinmin Optimization and Control Primary 49J53, 49J52, 47H05, Secondary 90C26, 47H09, 26B25 Level proximal subdifferential was introduced by Rockafellar recently for studying proximal mappings of possibly nonconvex functions. In this paper a systematic study of level proximal subdifferential is given. We characterize variational convexity of a function by local firm nonexpansiveness of proximal mappings or local relative monotonicity of level proximal subdifferential, and use them to study local convergence of proximal gradient method and others for variationally convex functions. Variational sufficiency guarantees that proximal gradient method converges to local minimizers rather than just critical points. We also investigate the existence, single-valuedness and integration of level proximal subdifferential, and quantify pointwise quadratic approximation (or Lipschitz smoothness) of a function. As a powerful tool, level proximal subdifferential provides deep insights into variational analysis and optimization. |
| title | Level proximal subdifferential, variational convexity, and pointwise quadratic approximation |
| topic | Optimization and Control Primary 49J53, 49J52, 47H05, Secondary 90C26, 47H09, 26B25 |
| url | https://arxiv.org/abs/2406.00648 |