On Carlier's inequality
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866913965318078464 |
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| author | Bauschke, Heinz H. Singh, Shambhavi Wang, Xianfu |
| author_facet | Bauschke, Heinz H. Singh, Shambhavi Wang, Xianfu |
| contents | The Fenchel-Young inequality is fundamental in Convex Analysis and Optimization. It states that the difference between certain function values of two vectors and their inner product is nonnegative. Recently, Carlier introduced a very nice sharpening of this inequality, providing a lower bound that depends on a positive parameter.
In this note, we expand on Carlier's inequality in three ways. First, a duality statement is provided. Secondly, we discuss asymptotic behaviour as the underlying parameter approaches zero or infinity. Thirdly, relying on cyclic monotonicity and associated Fitzpatrick functions, we present a lower bound that features an infinite series of squares of norms. Several examples illustrate our results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_14872 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On Carlier's inequality Bauschke, Heinz H. Singh, Shambhavi Wang, Xianfu Optimization and Control 26B25, 47H05 (Primary) 26D07, 90C25 (Secondary) The Fenchel-Young inequality is fundamental in Convex Analysis and Optimization. It states that the difference between certain function values of two vectors and their inner product is nonnegative. Recently, Carlier introduced a very nice sharpening of this inequality, providing a lower bound that depends on a positive parameter. In this note, we expand on Carlier's inequality in three ways. First, a duality statement is provided. Secondly, we discuss asymptotic behaviour as the underlying parameter approaches zero or infinity. Thirdly, relying on cyclic monotonicity and associated Fitzpatrick functions, we present a lower bound that features an infinite series of squares of norms. Several examples illustrate our results. |
| title | On Carlier's inequality |
| topic | Optimization and Control 26B25, 47H05 (Primary) 26D07, 90C25 (Secondary) |
| url | https://arxiv.org/abs/2206.14872 |