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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2025
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| Accès en ligne: | https://arxiv.org/abs/2504.14377 |
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| _version_ | 1866917342933417984 |
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| author | Rubbens, Anne Hendrickx, Julien M. Taylor, Adrien |
| author_facet | Rubbens, Anne Hendrickx, Julien M. Taylor, Adrien |
| contents | It is well known that functions (resp. operators) satisfying a property~$p$ on a subset $Q\subset \mathbb{R}^d$ cannot necessarily be extended to a function (resp. operator) satisfying~$p$ on the whole of~$\mathbb{R}^d$. Given $Q \subseteq \mathbb{R}^d$, this work considers the problem of obtaining necessary and ideally sufficient conditions to be satisfied by a function (resp. operator) on $Q$, ensuring the existence of an extension of this function (resp. operator) satisfying $p$ on $\mathbb{R}^d$.
More precisely, given some property $p$, we present a refinement procedure to obtain stronger necessary conditions to be imposed on $Q$. This procedure can be applied iteratively until the stronger conditions are also sufficient. We illustrate the procedure on a few examples, including the strengthening of existing descriptions for the classes of smooth functions satisfying a Łojasiewicz condition, convex blockwise smooth functions, Lipschitz monotone operators, strongly monotone cocoercive operators, and uniformly convex functions.
In most cases, these strengthened descriptions can be represented, or relaxed, to semi-definite constraints, which can be used to formulate tractable optimization problems on functions (resp. operators) within those classes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14377 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A constructive approach to strengthen algebraic descriptions of function and operator classes Rubbens, Anne Hendrickx, Julien M. Taylor, Adrien Optimization and Control It is well known that functions (resp. operators) satisfying a property~$p$ on a subset $Q\subset \mathbb{R}^d$ cannot necessarily be extended to a function (resp. operator) satisfying~$p$ on the whole of~$\mathbb{R}^d$. Given $Q \subseteq \mathbb{R}^d$, this work considers the problem of obtaining necessary and ideally sufficient conditions to be satisfied by a function (resp. operator) on $Q$, ensuring the existence of an extension of this function (resp. operator) satisfying $p$ on $\mathbb{R}^d$. More precisely, given some property $p$, we present a refinement procedure to obtain stronger necessary conditions to be imposed on $Q$. This procedure can be applied iteratively until the stronger conditions are also sufficient. We illustrate the procedure on a few examples, including the strengthening of existing descriptions for the classes of smooth functions satisfying a Łojasiewicz condition, convex blockwise smooth functions, Lipschitz monotone operators, strongly monotone cocoercive operators, and uniformly convex functions. In most cases, these strengthened descriptions can be represented, or relaxed, to semi-definite constraints, which can be used to formulate tractable optimization problems on functions (resp. operators) within those classes. |
| title | A constructive approach to strengthen algebraic descriptions of function and operator classes |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2504.14377 |