Zadeh’s Extension Principle for Continuous Functions of Non-Interactive Variables: A Parallel Optimization Approach

There is a growing interest in the use of fuzzy intervals in many engineering applications. However, a direct implementation of Zadeh's extension principle, which forms the basis for computing with fuzzy intervals, is still computationally too demanding for practical use. In the case of a continuous function and fuzzy intervals that describe non-interactive variables as inputs, the output is a fuzzy interval as well and can be determined for each α-cut separately. The problem, thus, reduces to finding the endpoints of these α-cuts, which amounts to a number of interwoven optimization problems. In the case of a non-monotone continuous function, however, these optimization problems are non-trivial. In this paper, different optimization algorithms are applied for that purpose: Gradient Descent based on Sequential Quadratic Programming, Simplex-Simulated Annealing, Particle Swarm Optimization, and Particle Swarm Optimization combined with Gradient Descent. In addition, two approaches are followed to determine a suitable number of α-cuts: either a fixed, predetermined number is used, or an initially (very) small number is chosen that is subsequently increased according to a linearity criterion. Both a non-parallel and a parallel implementation are designed. The parallel version is restricted to work with Particle Swarm Optimization and employs communication to optimize its (internal) performance by exploiting the dependence between the various optimization problems. Different configurations are evaluated on a set of benchmark functions in terms of the mean area under the output fuzzy interval and the number of function evaluations. Particle Swarm Optimization combined with Gradient Descent starting from a small number of α-cuts leads to the most accurate fuzzy intervals at the cost of a relatively large number of function evaluations.