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 W-gH-gradient efficient method for the optimization problems with interval-valued functions. The convergence analysis and the step-wise algorithms of both the methods are presented. It is observed that the 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 idea 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 are 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 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.

Generalized-Hukuhara Subgradient Method for Optimization Problem with Interval-valued Functions and its Application in Lasso Problem

“…In the field of optimization, to find out the best element or one of the best elements, an ordering is inevitably required. Since unlike the real numbers, intervals are not linearly ordered, most of the researches [5,6,14,16,29,30,33,34] developed the theories of IVFs based on partial ordering relations of intervals proposed by Isibuchi and Tanka [19]. In [3] some ordering relations based on the parametric representation of intervals are proposed.…”

“…Recently, Ghosh et al [14] have introduced the idea of gH-directional derivative, gH-Gâteaux derivative, and gH-Fréchet derivative of IVFs. Further, in [16], a new concept of gH-differentiability that is equipped with a linearity concept of IVFs is reported.…”

“…Since using parametric representations of the corresponding IVFs of IOPs or converting the IOPs into real-valued multiobjective optimization problems are practically quite difficult, recently, researchers (for instance, see [13,16,31], etc.) are trying to develop the theories and techniques to obtain the solutions to IOPs without using a real-valued multiobjective approach or parametric approach.…”

“…Stefanini and Arana-Jiménez [31] proposed KKT conditions and Ghosh et al [13] proposed generalized KKT conditions to obtain the solution of the IOPs. In [16], Ghosh et al developed a few gradient-descent techniques for smooth IOPs. However, a technique to deal with nonsmooth IOPs, without using realvalued multiobjective approach or parametric approach, is still undeveloped.…”

“…Theorem 2.1. (See [16]). Let an IVF F on a nonempty open convex subset X of R n be gH-differentiable at x ∈ X .…”