VMath matricese:optimize:distancee:torri_e
http://vmath.ru/vf5/
2026-10-13T04:36:47+0300VMath
http://vmath.ru/vf5/
http://vmath.ru/vf5/lib/tpl/vf5/images/favicon.icotext/html2023-07-14T22:59:30+0300Anonymous (anonymous@undisclosed.example.com)Inverse problem for the Fermat-Torricelli problem
http://vmath.ru/vf5/matricese/optimize/distancee/torri_e/inverse?rev=1689364770&do=diff
Author --- Русская версия
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Fermat-Torricelli problem and its development
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Inverse problem for the Fermat-Torricelli problem
Problem. Find the values for the
weights $ \{m_j\}_{j=1}^{K} $
with the aim for the corresponding objective function $ \sum_{j=1}^{K} m_j |PP_j| $ to posses a minimum point precisely at $ P_{\ast} \not\in \{P_j\}_{j=1}^K $.
Planar case
TheoremLet the vertices of the triangle$ P_{1}P_{2}P_3 $$$
m_1^{\ast} = |P_{\ast}P_1| \cdot \left|
\begin{arra…text/html2020-03-11T14:00:22+0300Anonymous (anonymous@undisclosed.example.com)Problem
http://vmath.ru/vf5/matricese/optimize/distancee/torri_e/pres020714?rev=1583924422&do=diff
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Problem
$$
F(P)=\frac{m_1}{|PP_1|}+\dots+ \frac{m_K}{|PP_K|}
$$
$$ \{P_1,\dots,P_K\} \subset \mathbb R^n,\ \{m_1,\dots,m_K\} \subset \mathbb R $$
$$ | \cdot | - \mbox{ Euclidean metrics } $$
$ \le (K-1)^2 $.
Difficulties
: $ n=2,\ K= 3 $;
$$ F(P)=\frac{m_1}{|PP_1|}+\frac{m_2}{|PP_2|}+\frac{m_3}{|PP_3|} $$
$$
\frac{D\, F}{D\, P} = \mathbb O
$$
$$
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\left\{\begin{array}{l}
\di…