Interval-valued Functions Research Articles

On preinvex interval-valued functions and unconstrained interval-valued optimization problems

The main objective of this paper is to investigate the generalized convexity of interval-valued functions under the total order relation and apply it to a class of unconstrained interval-valued optimization problems. For this purpose, we present the new definition of preinvex interval-valued functions and obtain its several fascinating characterizations. Then, we introduce the $\preceq_{cw}$-semicontinuity and discuss its relationship with preinvex interval-valued functions. As applications related to preinvex interval-valued functions, we study a class of unconstrained interval-valued optimization problems and discuss the existence theorem of its optimal solution.

Hermite–Hadamard-type Inequalities for hbar-preinvex Interval-Valued Functions via Fractional Integral

We present a comprehensive study on Hermite–Hadamard-type inequalities for interval-valued functions that are hbar-preinvex, using the Riemann–Liouville fractional integral. Our research extends and generalizes some existing results found in the literature. In addition, we provide accurate proofs for the main theorems originally derived by Srivastava et al. in their publication titled “Hermite–Hadamard Type Inequalities for Interval-Valued Preinvex Functions via Fractional Integral Operators" (Int. J. Comput. Int. Sys. 15(1):8, 2022). Finally, we illustrate our findings through a practical example to demonstrate the validity of our results.

Solving nonsmooth interval optimization problems based on interval-valued symmetric invexity

This paper focuses on a nonsmooth nonconvex interval-valued optimization problem. For this, we propose interval-valued symmetric invexity, interval-valued symmetric pseudo-invexity and interval-valued symmetric quasi-invexity in terms of the symmetric gH-differentiable interval-valued functions. Some important properties of these generalized convexities are also discussed. By utilizing these new concepts, we establish sufficient Karush–Kuhn–Tucker conditions for the considered problem. Further, the Wolfe and Mond–Weir type dual problems are associated and weak, strong and strict converse duality results have been derived. Finally, we apply the developed theory to a binary classification problem of interval data by support vector machine.

Some New Properties of Exponential Trigonometric Convex Functions Using up and down Relations over Fuzzy Numbers and Related Inequalities through Fuzzy Fractional Integral Operators Having Exponential Kernels

The concept of convexity is fundamental in order to produce various types of inequalities. Thus, convexity and integral inequality are closely related. The objectives of this paper are to present a new class of up and down convex fuzzy number valued functions known as up and down exponential trigonometric convex fuzzy number valued mappings (UDET-convex FNVMs) and, with the help of this newly defined class, Hermite–Hadamard-type inequalities (H–H-type inequalities) via fuzzy inclusion relation and fuzzy fractional integral operators having exponential kernels. This fuzzy inclusion relation is level-wise defined by the interval-based inclusion relation. Furthermore, we have shown that our findings apply to a significant class of both novel and well-known inequalities for UDET-convex FNVMs. The application of the theory developed in this study is illustrated with useful instances. Some very interesting examples are provided to discuss the validation of our main results. These results and other approaches may open up new avenues for modeling, interval-valued functions, and fuzzy optimization problems.

Gradient-based descent linesearch to solve interval-valued optimization problems under gH-differentiability with application to finance

In this article, gradient based descent line search scheme is proposed to solve interval optimization problems under generalized Hukuhara differentiability. The innovation and importance of these concepts are presented from practical and computational point of view. The necessary condition for existence of critical point is presented in inclusion form of interval-valued gradient, without transforming it into real valued function. Suitable efficient descent direction is chosen based on the monotonic property of the interval-valued function and specific interval ordering. Mathematical convergence of the scheme is proved under the assumption of Inexact line search for interval optimization problem. The theoretical developments are implemented with a set of interval test problems in different dimensions. A possible application in finance is provided and solved by our proposed scheme.

Inequalities for interval-valued Riemann diamond-alpha integrals

We propose the concept of Riemann diamond-alpha integrals for time scales interval-valued functions. We first give the definition and some properties of the interval Riemann diamond-alpha integral that are naturally investigated as an extension of interval Riemann nabla and delta integrals. With the help of the interval Riemann diamond-alpha integral, we present interval variants of Jensen inequalities for convex and concave interval-valued functions on an arbitrary time scale. Moreover, diamond-alpha Hölder’s and Minkowski’s interval inequalities are proved. Also, several numerical examples are provided in order to illustrate our main results.

Optimality conditions in a class of generalized convex optimization problems with the multiple interval-valued objective function

In this paper, a new class of generalized convex optimization problems with a multiple interval-valued objective function is considered. The concepts of B-preinvexity, B-invexity, and the generalized of B-invexity are extended to interval-valued functions. Additionally, several properties of interval-valued B-preinvex and B-invex functions are investigated. The Karush–Kuhn–Tucker necessary optimality conditions for a (weak LU-Pareto) LU-Pareto solution are established for a differentiable vector optimization problem with a multiple interval-valued objective function. Furthermore, the sufficient optimality conditions for both LU and UC order relations are derived for such interval-valued vector optimization problems under appropriate (generalized) B-invexity hypotheses. Suitable examples are provided to illustrate the aforementioned results.

GENERALIZED HERMITE–HADAMARD INCLUSIONS FOR A GENERALIZED FRACTIONAL INTEGRAL

We introduce new generalized fractional integrals for interval-valued functions. Then we prove generalized Hermite–Hadamard type inclusions for interval-valued convex functions using these newly defined generalized fractional integrals. We also show that these results generalize several results obtained in earlier works.

A Minimum Principle for Stochastic Optimal Control Problem with Interval Cost Function

In this paper, we study an optimal control problem in which their cost function is interval-valued. Also, a stochastic differential equation governs their state space. Moreover, we introduce a generalized version of Bellman's optimality principle for the stochastic system with an interval-valued cost function. Also, we obtain the Hamilton–Jacobi–Bellman equations and their control decisions. Two numerical examples happen in finance in which their cost function are interval-valued functions, illustrating the efficiency of the discussed results. The obtained results provide significantly reliable decisions compared to the case where the conventional cost function is applied.

Some New Versions of Fractional Inequalities for Exponential Trigonometric Convex Mappings via Ordered Relation on Interval-Valued Settings

This paper’s main goal is to introduce left and right exponential trigonometric convex interval-valued mappings and to go over some of their important characteristics. Additionally, we demonstrate the Hermite–Hadamard inequality for interval-valued functions by utilizing fractional integrals with exponential kernels. Moreover, we use the idea of left and right exponential trigonometric convex interval-valued mappings to show various findings for midpoint- and Pachpatte-type inequalities. Additionally, we show that the results provided in this paper are expansions of several of the results already demonstrated in prior publications The suggested research generates variants that are applicable for conducting in-depth analyses of fractal theory, optimization, and research challenges in several practical domains, such as computer science, quantum mechanics, and quantum physics.

Approximate Solution of Linear Interval Fuzzy Ordinary Differential Equations

Approximate solution of fuzzy initial value problem composed of interval fuzzy ordinary differential equation subject to initial conditions given as fuzzy intervals is the main objective of this work. The considered fuzzy initial value problem was given with Hukuhara definition of the difference and derivatives for interval-valued functions, in which two cases are considered depending on the comparison between the lower and upper functions of the solution derivative. The followed definition for presenting the fuzzy interval is generalized using trapezoidal fuzzy number, while the followed approximated method used in this work is the variational iteration method and its modification to increase the accuracy and the rate of convergence to the exact solution.

Some Important Classes of the Continuous and Complex Interval-Valued Functions

This paper presents some important classes of the continuous functions defined from the set of real numbers to the space of complex intervals. These function spaces have an algebraic structure named as a quasilinear space which is suggested by Aseev in 1986. In this work, we analysis the quasilinear structure on the classes of the continuous and complex interval-valued functions. Further, we show that these spaces are the normed Ω-spaces. Finally, we examine the dimension of these function spaces.

Bipolar interval-valued fuzzy set in graph and hypergraph settings

Graphs and hypergraphs are popular models for data structured representation. For example, traffic data, weather data, and animal skeleton data are all described by graph structures. Interval-valued fuzzy sets change the membership function of general fuzzy sets from single value functions to interval-valued functions, and thus describe the fuzzy attributes of things in terms of fuzzy intervals, which is more in line with the characteristics of fuzzy objectives. This paper aims to define the bipolar interval-valued fuzzy hypergraph to reveal the inner relationship of fuzzy data, and give some characterizations of it. The characteristics of bipolar interval-valued intuitionistic fuzzy hypergraph and bipolar interval-valued Pythagorean fuzzy hypergraph are studied. In addition, we discuss the characteristics of the bipolar interval-valued fuzzy threshold graph. Finally, some instances are presented as the applications of bipolar interval-valued fuzzy hypergraphs.

New Fractional Integral Inequalities Pertaining to Center-Radius (cr)-Ordered Convex Functions

In this work, we use the idea of interval-valued convex functions of Center-Radius (cr)-order to give fractional versions of Hermite–Hadamard inequality. The results are supported by some numerical estimations and graphical representations considering some suitable examples. The results are novel in the context of cr-convex interval-valued functions and deal with differintegrals of the p+s2 type. We believe this will be an important contribution to spurring additional research.

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 a 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-Gâteaux differentiable IVF, the proposed Ekeland’s variational principle is used.

On variational methods for interval-valued functions with some applications

In this paper, we introduce the concept of sequential weak lower semicontinuity of interval-valued functions defined on a Banach space. We focus on the direct methods of the calculus of variations for interval-valued functions and extend some well-known results of F.E. Browder from scalar results to interval-valued cases. Moreover, we obtain some new results on the Palais–Smale condition and the coercivity for interval-valued functions. Finally, we apply the obtained results to interval-valued integral functions, to interval-valued optimal control problems, and to variational problems for interval-valued functions.

Some variants on Mercer's Hermite-Hadamard like inclusions of interval-valued functions for strong Kernel

<abstract><p>Using Atangana-Baleanu ($ AB $) fractional integral operators, we first establish some fractional Hermite-Hadamard-Mercer inclusions for interval-valued functions in this study. In this, some fresh developments of the Hermite-Hadamard inequality for fractional integral operators are presented. A few instances are also given to support our conclusions. The resulting results could provide new insight into a variety of integral inequalities for fuzzy interval-valued functions, fractional interval-valued functions, and the optimization issues they raise. Finally, matrices-related applications are also shown.</p></abstract>

Some well known inequalities for $ (h_1, h_2) $-convex stochastic process via interval set inclusion relation

<abstract><p>This note introduces the concept of $ (h_1, h_2) $-convex stochastic processes using interval-valued functions. First we develop Hermite-Hadmard $ (\mathbb{H.H}) $ type inequalities, then we check the results for the product of two convex stochastic process mappings, and finally we develop Ostrowski and Jensen type inequalities for $ (h_1, h_2) $-convex stochastic process. Also, we have shown that this is a more generalized and larger class of convex stochastic processes with some remark. Furthermore, we validate our main findings by providing some non-trivial examples.</p></abstract>

On nondifferentiable semi-infinite multiobjective programming with interval-valued functions

In the paper, we investigate a class of nondifferentiable semi-infinite multiobjective programming problems such that all functions constituting them are interval-valued. We derive both Fritz John necessary optimality conditions and, under a constraint qualification, Karush-Kuhn-Tucker necessary optimality conditions for (weak) $ LU $-Pareto solutions in the considered nondifferentiable semi-infinite vector interval-valued optimization problem. Under appropriate invexity hypotheses, we also prove sufficient optimality conditions for this semi-infinite interval-valued vector optimization problem. Further, we also use the $ l_{1} $ exact function method for solving the aforesaid nondifferentiable interval-valued multicriteria optimization problem. Then, we analyze the property of exactness of the penalization for the absolute value exact penalty function method under assumption that the functions involved in the considered semi-infinite multiobjective programming problem are locally Lipschitz interval-valued invex functions. The conditions guaranteeing the equivalence of the sets of (weak) $ LU $-Pareto solutions in the original semi-infinite interval-valued multiobjective programming problem and its associated vector penalized optimization problem with the multiple interval-valued $ l_{1} $ exact penalty function are given.

Some new versions of Jensen, Schur and Hermite-Hadamard type inequalities for $ \left({p}, \mathfrak{J}\right) $-convex fuzzy-interval-valued functions

<abstract> <p>To create various kinds of inequalities, the idea of convexity is essential. Convexity and integral inequality hence have a significant link. This study's goals are to introduce a new class of generalized convex fuzzy-interval-valued functions (convex 𝘍𝘐𝘝𝘍s) which are known as $ \left(\mathfrak{p}, \mathfrak{J}\right) $-convex 𝘍𝘐𝘝𝘍s and to establish Jensen, Schur and Hermite-Hadamard type inequalities for $ \left(\mathfrak{p}, \mathfrak{J}\right) $-convex 𝘍𝘐𝘝𝘍s using fuzzy order relation. The Kulisch-Miranker order relation, which is based on interval space, is used to define this fuzzy order relation level-wise. Additionally, we have demonstrated that, as special examples, our conclusions encompass a sizable class of both new and well-known inequalities for $ \left(\mathfrak{p}, \mathfrak{J}\right) $-convex 𝘍𝘐𝘝𝘍s. We offer helpful examples that demonstrate the theory created in this study's application. These findings and various methods might point the way in new directions for modeling, interval-valued functions and fuzzy optimization issues.</p> </abstract>