Duality of Hoffman constants
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917195228905472 |
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| author | Pena, Javier F. Vera, Juan C. Zuluaga, Luis F. |
| author_facet | Pena, Javier F. Vera, Juan C. Zuluaga, Luis F. |
| contents | We show that a suitable Slater condition implies a duality inequality between the Hoffman constants of the following feasibility problems: $$ \begin{array}{r} Ax-b \in S\\ x \in R \end{array} \qquad\text{ and }\qquad \begin{array}{r} c-A^T y \in R^*\\ y \in S^*. \end{array} $$ where $A\in \mathbb{R}^{m\times n}$, and $R\subseteq \mathbb{R}^n$ and $S\subseteq \mathbb{R}^m$ are reference polyhedral cones, with respective dual cones $R^*\subseteq \mathbb{R}^n$ and $S^*\subseteq \mathbb{R}^m$. Our approach relies on an exact characterization of Hoffman constants and introduces a novel Hoffman duality inequality for polyhedral set-valued mappings. These two fundamental results also yield a striking identity between the Hoffman constants of box-constrained feasibility problems, which feature a similar primal-dual structure with a box and a linear subspace as reference sets. Additionally, we establish a surprising identity between the Hoffman constants of box-constrained feasibility problems and the chi condition measures for weighted least-squares problems |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_09858 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Duality of Hoffman constants Pena, Javier F. Vera, Juan C. Zuluaga, Luis F. Optimization and Control 90C05, 90C25, 90C57 We show that a suitable Slater condition implies a duality inequality between the Hoffman constants of the following feasibility problems: $$ \begin{array}{r} Ax-b \in S\\ x \in R \end{array} \qquad\text{ and }\qquad \begin{array}{r} c-A^T y \in R^*\\ y \in S^*. \end{array} $$ where $A\in \mathbb{R}^{m\times n}$, and $R\subseteq \mathbb{R}^n$ and $S\subseteq \mathbb{R}^m$ are reference polyhedral cones, with respective dual cones $R^*\subseteq \mathbb{R}^n$ and $S^*\subseteq \mathbb{R}^m$. Our approach relies on an exact characterization of Hoffman constants and introduces a novel Hoffman duality inequality for polyhedral set-valued mappings. These two fundamental results also yield a striking identity between the Hoffman constants of box-constrained feasibility problems, which feature a similar primal-dual structure with a box and a linear subspace as reference sets. Additionally, we establish a surprising identity between the Hoffman constants of box-constrained feasibility problems and the chi condition measures for weighted least-squares problems |
| title | Duality of Hoffman constants |
| topic | Optimization and Control 90C05, 90C25, 90C57 |
| url | https://arxiv.org/abs/2312.09858 |