Global convergence of Karnik–Mendel algorithms

Karnik–Mendel (KM) algorithms are the most commonly used iterative type reduction methods in interval type-2 fuzzy sets and systems, as well as new techniques for computing the fuzzy weighted average (FWA). Various extensions and improvements have been proposed. However, no proof has been provided for the convergence of these extensions. It is necessary to provide the proof because many of the iterative algorithms may have divergence cases. In the present study, we provide a theoretical proof that KM algorithms exhibit global convergence. Different initialization methods and iteration formats can always obtain the same unique optimal solution. Thus, there are no concerns about the possibility of divergence in extensions of KM algorithms. Our proof provides theoretical support for the applications of KM algorithms, especially the type reduction designs used in type-2 fuzzy systems and FWA computations because of the important roles of KM algorithms in these methods.