Newton method to obtain efficient solutions of the optimization problems with interval-valued objective functions

Abstract

In this paper, a Newton method is proposed to obtain efficient solutions for the optimization problems with interval-valued objective functions. In the concept of efficient solution of the problem, a suitable partial ordering for a pair of intervals is used. Through the notion of generalized Hukuhara difference of a pair of intervals, the generalized Hukuhara differentiability of multi-variable interval-valued functions is defined and analyzed to develop the proposed method. The objective function in the problem is assumed to be twice continuously generalized Hukuhara differentiable. Under this hypothesis, it is exhibited that the method has a local quadratic rate of convergence. In order to improve the local convergence of the method to a global convergence, an updated Newton method is also given. The sequential algorithms and the convergence results of the proposed methods are demonstrated. Several numerical examples are presented to illustrate the proposed methodologies.