Tuning fuzzy control rules by the alpha constrained method which solves constrained nonlinear optimization problems

Learning of fuzzy control rules can be considered as a constrained nonlinear optimization problem, in which the objective function isn't differentiable. In this case, the problem can be solved by the combination of direct search method and penalty function method. However, it is difficult to know how much a candidate solution satisfies the constraints. We introduce the satisfaction level of constraints, similar to fuzzy constraints in fuzzy programming. The satisfaction level is a function which is 1 when the constraints are satisfied perfectly and approaches 0 as the satisfaction level becomes low. By using this level, we propose the α level comparison which compares the candidates based on the satisfaction level of constraints. We also propose the α constraint method which converts constrained problems to unconstrained problems using the α level comparison. Through some examples and the learning of fuzzy control rules, we show that the feasible solution can be obtained easily with our method while confirming the satisfaction level.