In this article, we study the approximate solutions set $\Lambda_b$ of an
inconsistent system of $\max-\min$ fuzzy relational equations $(S): A
\Box_{\min}^{\max}x =b$. Using the $L_\infty$ norm, we compute by an explicit
analytical formula the Chebyshev distance $\Delta~=~\inf_{c \in \mathcal{C}}
\Vert b -c \Vert$, where $\mathcal{C}$ is the set of second members of the
consistent systems defined with the same matrix $A$. We study the set
$\mathcal{C}_b$ of Chebyshev approximations of the second member $b$ i.e.,
vectors $c \in \mathcal{C}$ such that $\Vert b -c \Vert = \Delta$, which is
associated to the approximate solutions set $\Lambda_b$ in the following sense:
an element of the set $\Lambda_b$ is a solution vector $x^\ast$ of a system $A
\Box_{\min}^{\max}x =c$ where $c \in \mathcal{C}_b$. As main results, we
describe both the structure of the set $\Lambda_b$ and that of the set
$\mathcal{C}_b$. We then introduce a paradigm for $\max-\min$ learning weight
matrices that relates input and output data from training data. The learning
error is expressed in terms of the $L_\infty$ norm. We compute by an explicit
formula the minimal value of the learning error according to the training data.
We give a method to construct weight matrices whose learning error is minimal,
that we call approximate weight matrices.
Finally, as an application of our results, we show how to learn approximately
the rule parameters of a possibilistic rule-based system according to multiple
training data.

研究了模糊关系方程的近似解集和学习权重矩阵，提出了一种近似权重矩阵的方法，并应用于可能性规则系统的参数学习。