A Fuzzy gH-Finite Difference Newton Method for Solving Fuzzy Nonlinear Optimization Problems

Abstract

This paper introduces a new numerical technique for solving nonlinear fuzzy programs. This novel approach is of the Newton type and is founded on the gH-finite difference of fuzzy nonlinear functions. It finds a non-dominated solution to a multivariate, nonlinear, fuzzy optimization problem by combining the gH-difference concept for fuzzy quantities, the finite difference notion, and Newton’s algorithm. The process begins with the formulation of a finite difference gH of multivariate fuzzy nonlinear functions via gH-differentiability. This new method has the unique ability to efficiently handle optimization problems involving fuzzy nonlinear functions with highly complex derivatives. Unlike the fuzzy Newton methods presented in the literature, the method presented in this paper does not use the notion of derivative. Rather, it uses the fuzzy gH-finite difference technique to approximate the gradient and Hessian matrix of nonlinear fuzzy functions. Therefore, it applies to fuzzy nonlinear optimization problems for which the first and second partial gH-derivatives are difficult to obtain through classical calculation.