Estimation of fuzzy relational matrix by using probabilistic descent method

Fuzzy relation (and fuzzy relational matrices, in particular) and a class of fuzzy relational systems become important in many areas of applications like diagnosis of complex systems, pattern classification or control. The essential task emerging there requires determination of the numerical values of the fuzzy relations. In this paper we will look at this as a certain optimization problem and examine the use of a probabilistic descent method. Especially we will be concerned with the max-min composition and its numerical representation that legitimizes the use of the abovementioned algorithm. We will propose several approximations to the max and min functions that will allow us to cope with their nondifferentiable characters. These modifications pertain both to the functions as well as their derivatives. The results of extensive numerical studies are also reported.