Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2406.08513 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866913389703331840 |
|---|---|
| author | Gzyl, Henryk |
| author_facet | Gzyl, Henryk |
| contents | When there are no constraints upon the solutions of the equation $\mathbf{A}\mathbfξ= \mathbf{y},$ where $\mathbf{A}$ is a $K\times N-$matrix, $\mathbfξ\in\mathbb{R}^N$ and $\mathbf{y}\in\mathbb{R}^K$ a given vector, the description of the set of solutions as $\mathbf{y}$ varies in $\mathbb{R}^K$ is well known. But this is not so when the solutions are required to satisfy $\mathbfξ \in \mathcal{K}\prod_{i\leq j\leq N} [a_j,b_j],$ for finite $a_j\leq b_j: 1\leq j\leq N.$ Here we provide a description of the set of solutions as a surface in the constraint set, parameterized by the Lagrange multipliers that come up in a related optimization problem in which $\mathbf{A}\mathbfξ = \mathbf{y}$ appears as a constraint. It is the dependence of the Lagrange multipliers on the data vector $\mathbf{y}$ that determines how the solution changes as the datum changes. The geometry on the solutions is inherited from a Riemannian geometry on the set of constraints induced by the Hessian of an entropy of the Fermi-Dirac type which is the objective in the restatement of the optimization problem mentioned above. We prove that the set of solutions is contained in $\ker(\mathbf{A})^\perp$ in the metric defined as the Hessian of the entropy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_08513 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A geometry in the set of solutions to ill-posed linear problems with box constraints: Applications to probabilities on discrete sets Gzyl, Henryk Rings and Algebras Optimization and Control When there are no constraints upon the solutions of the equation $\mathbf{A}\mathbfξ= \mathbf{y},$ where $\mathbf{A}$ is a $K\times N-$matrix, $\mathbfξ\in\mathbb{R}^N$ and $\mathbf{y}\in\mathbb{R}^K$ a given vector, the description of the set of solutions as $\mathbf{y}$ varies in $\mathbb{R}^K$ is well known. But this is not so when the solutions are required to satisfy $\mathbfξ \in \mathcal{K}\prod_{i\leq j\leq N} [a_j,b_j],$ for finite $a_j\leq b_j: 1\leq j\leq N.$ Here we provide a description of the set of solutions as a surface in the constraint set, parameterized by the Lagrange multipliers that come up in a related optimization problem in which $\mathbf{A}\mathbfξ = \mathbf{y}$ appears as a constraint. It is the dependence of the Lagrange multipliers on the data vector $\mathbf{y}$ that determines how the solution changes as the datum changes. The geometry on the solutions is inherited from a Riemannian geometry on the set of constraints induced by the Hessian of an entropy of the Fermi-Dirac type which is the objective in the restatement of the optimization problem mentioned above. We prove that the set of solutions is contained in $\ker(\mathbf{A})^\perp$ in the metric defined as the Hessian of the entropy. |
| title | A geometry in the set of solutions to ill-posed linear problems with box constraints: Applications to probabilities on discrete sets |
| topic | Rings and Algebras Optimization and Control |
| url | https://arxiv.org/abs/2406.08513 |