Generalized-Hukuhara-Gradient Efficient-Direction Method to Solve Optimization Problems with Interval-Valued Functions and Its Application in Least-Squares Problems

Abstract

This article proposes a general gH-gradient efficient-direction method and a \({\mathcal{W}}\)-gH-gradient efficient method for the optimization problems with interval-valued functions. The convergence analysis and the step-wise algorithms of both methods are presented. It is observed that the \({\mathcal{W}}\)-gH-gradient efficient method converges linearly for a strongly convex interval-valued objective function. To develop the proposed methods and to study their convergence, the ideas of strong convexity and sequential criteria for gH-continuity of interval-valued function are illustrated. In the sequel, a new definition of gH-differentiability for interval-valued functions is also proposed. The new definition of gH-differentiability is described with the help of a newly defined concept of linear interval-valued function. It is noticed that the proposed gH-differentiability is superior to the existing ones. For a gH-differentiable interval-valued function, the relation of convexity with the gH-gradient of an interval-valued function and an optimality condition of an interval optimization problem is derived. For the derived optimality condition, a notion of efficient direction for interval-valued functions is introduced. The idea of efficient direction is used to develop the proposed gradient methods. As an application of the proposed methods, the least-square problem for interval-valued data by \({\mathcal{W}}\)-gH-gradient efficient method is solved. The proposed method for least square problems is illustrated by a polynomial fitting and a logistic curve fitting.