arXiv subjects

Amit Kumar Debnath

Publications and source records attributed to Amit Kumar Debnath.

4 recordsLinked to original sources

arXiv2021

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

In this study, a \emph{$gH$-subgradient technique} is developed to obtain efficient solutions to the optimization problems with nonsmooth nonlinear convex interval-valued functions. The algorithmic implementation of the developed $gH$-subgradient technique is illustrated. As an application of the proposed \emph{$gH$-subgradient technique}, an $\ell_1$ penalized linear regression problem, known as a \emph{lasso problem}, with interval-valued features is solved.

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arXiv2021

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

In this article, the concepts of gH-subgradients and gH-subdifferentials of interval-valued functions are illustrated. Several important characteristics of the gH-subdifferential of a convex interval-valued function, e.g., closeness, boundedness, chain rule, etc. are studied. Alongside, we prove that gH-subdifferential of a gH-differentiable convex interval-valued function only contains gH-gradient of that interval-valued function. It is observed that the gH-directional derivative of a convex interval-valued function in each direction is maximum of all the products of gH-subgradients and the direction. Importantly, we show that a convex interval-valued function is gH-Lipschitz continuous if it has gH-subgradients at each point in its domain. Furthermore, the relations between efficient solutions of an optimization problem with interval-valued function and its gH-subgradients are derived.

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arXiv2020

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

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.

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arXiv2020

Generalized Hukuhara-Clarke Derivative of Interval-valued Functions and its Properties

In this article, the notion of gH-Clarke derivative for interval-valued functions is proposed. To define the concept of gH-Clarke derivatives, the concepts of limit superior, limit inferior, and sublinear interval-valued functions are studied in the sequel. The upper gH-Clarke derivative of a gH-Lipschitz interval-valued function (IVF) is observed to be a sublinear IVF. It is found that every gH-Lipschitz continuous function is upper gH-Clarke differentiable. For a convex and gH-Lipschitz IVF, it is shown that the upper gH-Clarke derivative coincides with the gH-directional derivative. The entire study is supported by suitable illustrative examples.

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