arXiv subjects

Debdas Ghosh

Publications and source records attributed to Debdas Ghosh.

29 records · Page 2Linked to original sources

arXiv2023

Fritz-John optimality condition in fuzzy optimization problems and its application to classification of fuzzy data

The main objective of this paper is to derive the optimality conditions for one type of fuzzy optimization problems. At the beginning, we define a cone of descent direction for fuzzy optimization, and prove that its intersection with the cone of feasible directions at an optimal point is an empty set. Then, we present first-order optimality conditions for fuzzy optimization problems. Furthermore, we generalize the Gordan's theorem for fuzzy linear inequality systems and utilize it to deduce the Fritz-John optimality condition for the fuzzy optimization with inequality constraints. Finally, we apply the optimality conditions established in this paper to a binary classification problem for support vector machines with fuzzy data. In the meantime, numerical examples are described to demonstrate the primary findings proposed in the present paper.

math.OC↗

arXiv2022

Generalized Hukuhara Weak Subdifferential and its Application on Identifying Optimality Conditions for Nonsmooth Interval-valued Functions

In this article, we introduce the idea of $gH$-weak subdifferential for interval-valued functions (IVFs) and show how to calculate $gH$-weak subgradients. It is observed that a nonempty $gH$-weak subdifferential set is closed and convex. In characterizing the class of functions for which the $gH$-weak subdifferential set is nonempty, it is identified that this class is the collection of $gH$-lower Lipschitz IVFs. In checking the validity of sum rule of $gH$-weak subdifferential for a pair of IVFs, a counterexample is obtained, which reflects that the sum rule does not hold. However, under a mild restriction on one of the IVFs, one-sided inclusion for the sum rule holds. Next, as applications, we employ $gH$-weak subdifferential to provide a few optimality conditions for nonsmooth IVFs. Further, a necessary optimality condition for interval optimization problems with difference of two nonsmooth IVFs as objective is established. Lastly, a necessary and sufficient condition via augmented normal cone and $gH$-weak subdifferential of IVFs for finding weak efficient point is presented.

math.OC↗

arXiv2022

Generalized Hukuhara Hadamard Derivative of Interval-valued Functions and Its Applications to Interval Optimization

In this article, we study the notion of gH-Hadamard derivative for interval-valued functions (IVFs) and its applications to interval optimization problems (IOPs). It is shown that the existence of gH-Hadamard derivative implies the existence of gH-Frechet derivative and vise-versa. Further, it is proved that the existence of gH-Hadamard derivative implies the existence of gH-continuity of IVFs. We found that the composition of a Hadamard differentiable real-valued function and a gH-Hadamard differentiable IVF is gHHadamard differentiable. Further, for finite comparable IVF, we prove that the gH-Hadamard derivative of the maximum of all finite comparable IVFs is the maximum of their gH-Hadamard derivative. The proposed derivative is observed to be useful to check the convexity of an IVF and to characterize efficient points of an optimization problem with IVF. For a convex IVF, we prove that if at a point the gH-Hadamard derivative does not dominate to zero, then the point is an efficient point. Further, it is proved that at an efficient point, the gH-Hadamard derivative does not dominate zero and also contains zero. For constraint IOPs, we prove an extended Karush-Kuhn-Tucker condition by using the proposed derivative. The entire study is supported by suitable examples.

math.OC↗

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.

math.OC↗

arXiv2021

Generalized-Hukuhara Subdifferential Analysis and Its Application in Nonconvex Composite Optimization Problems with Interval-valued Functions

In this article, we study $gH$-subdifferential calculus of convex interval-valued functions (IVFs) and apply it in a nonconvex composite model of interval optimization problems (IOPs). It is found that the $gH$-directional derivative of maximum of finitely many comparable IVFs is the maximum of their $gH$-directional derivative. Proposed concepts of $gH$-subdifferential are observed to be useful to derive Fritz-John-type and KKT-type efficiency conditions for weak efficient solutions of IOPs. Further, we extract a necessary and sufficient condition to characterize the weak efficient solutions of nonconvex composite IOPs by applying the proposed concepts. To derive the results on $gH$-subdifferentials, the concepts of limit supremum and limit infimum with certain properties for IVFs are defined in the sequel. The whole analysis is supported by appropriate expository examples.

math.OC↗

arXiv2021

Weak sharp minima for interval-valued functions and its primal-dual characterizations using generalized Hukuhara subdifferentiability

This article introduces the concept of weak sharp minima (WSM) for convex interval-valued functions (IVFs). To identify a set of WSM of a convex IVF, we provide its primal and dual characterizations. The primal characterization is given in terms of $gH$-directional derivatives. On the other hand, to derive dual characterizations, we propose the notions of the support function of a subset of $I(\mathbb{R})^{n}$ and $gH$-subdifferentiability for convex IVFs. Further, we develop the required $gH$-subdifferential calculus for convex IVFs. Thereafter, by using the proposed $gH$-subdifferential calculus, we provide dual characterizations for the set of WSM of convex IVFs.

math.OC↗

arXiv2021

Ekeland's Variational Principle for Interval-valued Functions

In this paper, we attempt to propose Ekeland's variational principle for interval-valued functions (IVFs). To develop the variational principle, we study the concept of sequence of intervals. In the sequel, the idea of gH-semicontinuity for IVFs is explored. A necessary and sufficient condition for an IVF to be gH-continuous in terms of gH-lower and upper semicontinuity is given. Moreover, we prove a characterization for gH-lower semicontinuity by the level sets of the IVF. With the help of this characterization result, we ensure the existence of a minimum for an extended gH-lower semicontinuous, level-bounded and proper IVF. To find an approximate minima of a gH-lower semicontinuous and gH-Gateaux differentiable IVF, the proposed Ekeland's variational principle is used.

math.OC↗

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.

math.OC↗

arXiv2019

Globalized Robust Markov Perfect Equilibrium for Discounted Stochastic Games and its Application on Intrusion Detection in Wireless Sensor Networks: Part I -- Theory

In this article, we study a discounted stochastic game to model resource optimal intrusion detection in wireless sensor networks. To address the problem of uncertainties in various network parameters, we propose a globalized robust game theoretic framework for discounted robust stochastic games. A robust solution to the considered problem is an optimal point that is feasible for all realizations of data from a given uncertainty set. To allow a controlled violation of the constraints when the parameters move out of the uncertainty set, the concept of globalized robust framework comes into view. In this article, we formulate a globalized robust counterpart for the discounted stochastic game under consideration. With the help of globalized robust optimization, a concept of globalized robust Markov perfect equilibrium is introduced. The existence of such an equilibrium is shown for a discounted stochastic game when the number of actions of the players is finite. The contraction mapping theorem, Kakutani fixed point theorem and the concept of equicontinuity are used to prove the existence result. To compute a globalized robust Markov perfect equilibrium for the considered discounted stochastic game, a tractable representation of the proposed globalized robust counterpart is also provided. Using the derived tractable representation, we formulate a globalized robust intrusion detection system for wireless sensor networks. The simulation result reveals that the proposed globalized solution is much less sensitive to data perturbations than the robust solution.

cs.GT↗

arXiv2019

Differentiability and optimality of a fuzzy function

In this article, we introduce a differentiability concept for fuzzy functions $\tilde{f}: F(\mathbb{R}) \to F(\mathbb{R})$, where $F(\mathbb{R})$ is the set of all fuzzy numbers. With the help of the proposed differentiability notion, we prove a necessary and sufficient condition for optimality to obtain a non-dominated solution of a fuzzy optimization problem.

math.OC↗

arXiv2019

Vector Contraction Analysis for Nonlinear Dynamical Systems

This paper derives new results for the analysis of nonlinear systems by extending contraction theory in the framework of vector distances. A new tool, vector contraction analysis utilizing a notion of the vector-valued norm which evidently induces a vector distance between any pair of trajectories of the system, offers an amenable framework as each component of vector-valued norm function satisfies fewer strict conditions as that of standard contraction analysis. Particularly, every element of vector-valued norm derivative need not be strictly negative definite for convergence of any pair of trajectories of the system. Moreover, vector-valued norm derivative satisfies a component-wise inequality employing some comparison system. In fact, the convergence analysis is performed by comparing the relative distances between any pair of the trajectories of the original nonlinear system and the comparison system. Comparison results are derived by utilizing the concepts of quasi-monotonicity property of the function and vector differential inequalities. Moreover, the results are also derived in the framework of the cone ordering instead of utilizing component-wise inequalities between vectors. In addition, the proposed framework is illustrated by examples.

math.OC↗