Trapezoidal approximations of fuzzy numbers using quadratic programs

Abstract

In this paper we will prove that the nearest trapezoidal approximation of fuzzy numbers with respect to weighted L2-type metrics with or without additional constraints can be obtained via quadratic programs. Actually, the approach is even more general based on so called finite polyhedral subsets of fuzzy numbers which include most of the important special classes of fuzzy numbers available in the literature. In particular, we will recapture the algorithm to compute the nearest weighted trapezoidal approximation of a fuzzy number by a method which we believe that has the potential to be extended to more complex approximation problems. Then, we will improve the Lipschitz constant of the trapezoidal approximation operator preserving the ambiguity. To achieve this improved result we will exploit the fact that we have an analytical expression for this operator. However, note that the same result is obtained if this solution function is described by quadratic programs. Therefore, for similar problems we still can obtain Lipschitz constants for the approximation operator even if an analytical expression of this operator is not available.