A Newton method for capturing efficient solutions of interval optimization problems

Abstract

In this article, we propose a Newton method to obtain an efficient solution for interval optimization problems. In the concept of an efficient solution of the problem, a suitable partial ordering for a pair of intervals is used. The notion of generalized Hukuhara difference of intervals, and hence, the generalized Hukuhara differentiability of multi-variable interval-valued functions is 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 shown 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 proposed. The sequential algorithms and the convergence results of the proposed methods are demonstrated. The methodologies are illustrated with suitable numerical examples.