Penalty Fuzzy Function for Derivative-Free Optimization

Penalty and Barrier methods are normally used to solve Nonlinear Optimization Constrained Problems. The problems appear in areas such as engineering and are often characterized by the fact that involved functions (objective and constraints) are non-smooth and/or their derivatives are not know. This means that optimization methods based on derivatives cannot be used. A Java based API was implemented, including only derivative-free optimization methods, to solve both constrained and unconstrained problems, which includes Penalty and Barriers methods. In this work a new penalty function, based on Fuzzy Logic, is presented. This function imposes a progressive penalization to solutions that violate the constraints. This means that the function imposes a low penalization when the violation of the constraints is low and a heavy penalization when the violation is high. The value of the penalization is not known in beforehand, it is the outcome of a fuzzy inference engine. Numerical results comparing the proposed function with two of the classic penalty/barrier functions are presented. Regarding the presented results one can conclude that the proposed penalty function besides being very robust also exhibits a very good performance.