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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2501.01854 |
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| _version_ | 1866913634522759168 |
|---|---|
| author | Mohan, Shravan |
| author_facet | Mohan, Shravan |
| contents | In this brief note, it is shown that the function p^TW log(p) is convex in p if W is a diagonally dominant positive definite M-matrix. The techniques used to prove convexity are well-known in linear algebra and essentially involves factoring the Hessian in a way that is amenable to martix analysis. Using similar techniques, two classes of convex homogeneous polynomials is derived - namely, p^TW p2 and (p^k)^TW p^k - the latter also happen to be SOS-convex. Lastly, usign the same techniques, it is also shown that the function p^TW ep is convex over the positive reals only if W is a non-negative diagonal matrix. Discussions regarding the utility of these functions and examples accompany the results presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01854 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On extending the class of convex functions Mohan, Shravan Optimization and Control In this brief note, it is shown that the function p^TW log(p) is convex in p if W is a diagonally dominant positive definite M-matrix. The techniques used to prove convexity are well-known in linear algebra and essentially involves factoring the Hessian in a way that is amenable to martix analysis. Using similar techniques, two classes of convex homogeneous polynomials is derived - namely, p^TW p2 and (p^k)^TW p^k - the latter also happen to be SOS-convex. Lastly, usign the same techniques, it is also shown that the function p^TW ep is convex over the positive reals only if W is a non-negative diagonal matrix. Discussions regarding the utility of these functions and examples accompany the results presented. |
| title | On extending the class of convex functions |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2501.01854 |