Differential Evolution Methods for LU-fuzzy Arithmetic

The paper illustrates a di¤erential evolution (DE) algorithm to calculate the level-cuts of the fuzzy extension of a multidimensional real valued function to fuzzy numbers. The method, in the frame of the LU parametrization, decomposes the fuzzy extension engine into a set of "nested" min and max boxconstrained optimization problems that can be solved by appropriate forms of the DE algorithm, i.e. based on multi populations which cooperate during the search phase and specialize to …nd the global min (corresponding to Lower branch) and the global max (Upper branch), both gaining e¢ ciency from the work done for a level-cut to the subsequent ones. A special version of the algorithm is designed to the case of di¤erentiable functions, for which a representation of the fuzzy numbers is used to improve e¢ ciency and quality of calculations. Some preliminary results indicate DE methods as promising tools: in a number of test problems, its computational complexity grows on average superlinearly (of degree less than 1.5) in the number of variables of the function to be extended.