Interval-valued Objective Function Research Articles

First and second order necessary optimality conditions for multiobjective programming with interval-valued objective functions on Riemannian manifolds

The growing dependence on optimization models in decision-making has created a demand for tools that can facilitate the formulation and resolution of a broader range of real-world processes and systems associated with human activity. These situations often involve assumptions that diverge from traditional optimization methodologies. One viable approach for addressing optimization problems in real-life scenarios with uncertainty is interval-valued optimization. Taking into account the significance of interval-valued optimization, in this paper, we derive first and second order necessary optimality conditions for a multi-objective programming problem with interval-valued objective functions defined on a Riemannian manifold. To establish these conditions, we consider the objective functions to be weakly differentiable and twice weakly differentiable for first and second order, respectively. Additionally, we assume that the constraints, both equality and inequality constraints, are differentiable and twice differentiable for first and second order conditions respectively. The first order as well as second order necessary conditions are derived under two types of constraint qualifications. Furthermore, we provide illustrative examples to demonstrate the application of the established results.

Second-Order Optimality Conditions for Interval-Valued Optimization Problem

In this paper, we discuss the second-order KKT-type optimality conditions for interval-valued optimization problem. To derive second-order necessary and sufficient conditions, we use two different methodologies: square slack variable method and the theory of geometric cone optimality conditions for interval-valued objective functions. Furthermore, we discuss two types of order relations LU and center–radius (C–R). To show the validity of our results, we have demonstrated some nontriv ial examples.

Second-order necessary and sufficient optimality conditions for multiobjective interval-valued nonlinear programming

In this article, the second-order optimality conditions for nonlinear programming with multiple interval-valued objective functions (in brief, NPMIOF) are studied. In the first part, the second- and first-order necessary optimality conditions for some types of efficient solutions of NPMIOF are established under of Abadie second- and first-order constraint qualifications. In the next part, we investigate both primal and dual second-order sufficient optimality conditions. Our second-order necessary conditions enhance the previous results. The second-order sufficient optimality conditions are new.

Impact of Carbon Emissions and Advance Payment on Optimal Decisions for Perishable Products via Parametric Approach of Interval

Under the heading of sustainability challenges, this study incorporates payment procedures and inventory selections. The article’s goal is to acquire insight into how the methods of payment affect perishable product inventory selections under the widely utilised carbon tax regime in view of policies to reduce emissions. Uncertainty arises as a natural consequence of the unpredictable behaviour of customers. Keeping these impacts in mind, an interval-valued inventory model is introduced, where all the related inventory parameters are considered interval-valued. In addition, the vendor offers an interval-valued discount rate to customers against advance payment. Due to the inventory parameters being chosen as interval-valued, the objective function is changed into an interval-valued form, and the corresponding differential equation is changed into an interval differential equation. To solve the interval-valued differential equation, a parametric approach to interval is introduced, and the corresponding interval-valued objective function is constructed. Interval order relations and the MATHEMATICA software are used to solve the objective function with interval values. To assess the validity of the proposed model, one numerical example is solved, and a sensitivity analysis of the ideal course of action is performed.

On first and second order multiobjective programming with interval-valued objective functions

The growing use of optimization models to help decision making has created a demand for such tools that allow formulating and solving more models of real-world processes and systems related to human activity in which hypotheses are not verify in a way specific for classical optimization. One of the approaches for real-world extremum problems under uncertainty is interval-valued optimization. In this paper, a twice differentiable vector optimization problem with multiple interval-valued objective function and both inequality and equality constraints is considered. In this paper, the first order necessary optimality conditions of Karush-Kuhn-Tucker type are proved for differentiable interval-valued vector optimization problems under the first order constraint qualification. If the interval-valued objective function is assumed to be twice weakly differentiable and constraints functions are assumed to be twice differentiable, then two types of second order necessary optimality conditions under two various constraint qualifications are proved for such smooth interval-valued vector optimization problems. Finally, in order to illustrate the Karush-Kuhn-Tucker type necessary optimality conditions established in the paper, an example of an interval-valued optimization is given.

Semi-infinite interval equilibrium problems: optimality conditions and existence results

This paper aims to obtain new Karush–Kuhn–Tucker optimality conditions for solutions to semi-infinite interval equilibrium problems with interval-valued objective functions. The Karush–Kuhn–Tucker conditions for the semi-infinite interval programming problem are particular cases of those found in this paper for constrained equilibrium problem. We illustrate this with some examples. In addition, we obtain solutions to the interval equilibrium problem in the unconstrained case. The results presented in this paper extend the corresponding results in the literature.

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.

A subgradient-based neurodynamic algorithm to constrained nonsmooth nonconvex interval-valued optimization

In this paper, a subgradient-based neurodynamic algorithm is presented to solve the nonsmooth nonconvex interval-valued optimization problem with both partial order and linear equality constraints, where the interval-valued objective function is nonconvex, and interval-valued partial order constraint functions are convex. The designed neurodynamic system is constructed by a differential inclusion with upper semicontinuous right-hand side, whose calculation load is reduced by relieving penalty parameters estimation and complex matrix inversion. Based on nonsmooth analysis and the extension theorem of the solution of differential inclusion, it is obtained that the global existence and boundedness of state solution of neurodynamic system, as well as the asymptotic convergence of state solution to the feasible region and the set of LU-critical points of interval-valued nonconvex optimization problem. Several numerical experiments and the applications to emergency supplies distribution and nondeterministic fractional continuous static games are solved to illustrate the applicability of the proposed neurodynamic algorithm.

Necessary and sufficient optimality conditions for fractional interval-valued variational problems

In this paper a special kind of variational programming problem involving fractional interval-valued objective function is considered. For such type of problem, insights into LU optimal solutions have been discussed. Using the LU optimal concept, we established optimality conditions for the considered problem. Further, We formulated a Mond-Weir dual problem and discussed appropriate duality theorems for the relationship between dual and primal problems.

Optimality conditions and duality for continuous-time programming with multiple interval-valued objective functions

Optimality conditions and duality for continuous-time programming with multiple interval-valued objective functions

Optimality conditions for nonlinear optimization problems with interval-valued objective function in admissible orders

Optimality conditions for nonlinear optimization problems with interval-valued objective function in admissible orders

On approximate quasi Pareto solutions in nonsmooth semi-infinite interval-valued vector optimization problems

This paper deals with approximate solutions of a nonsmooth semi-infinite programming with multiple interval-valued objective functions. We first introduce four types of approximate quasi Pareto solutions of the considered problem by considering the lower-upper interval order relation and then apply some advanced tools of variational analysis and generalized differentiation to establish necessary optimality conditions for these approximate solutions. Sufficient conditions for approximate quasi Pareto solutions of such a problem are also provided by means of introducing the concepts of approximate (strictly) pseudo-quasi generalized convex functions defined in terms of the limiting subdifferential of locally Lipschitz functions. Finally, a Mond–Weir type dual model in approximate form is formulated, and weak, strong and converse-like duality relations are proposed.

Optimization of the mathematical programming and applications

The present investigation responds to the need to solve optimization problems with optimality conditions. The KKT conditions are considered for multiobjective optimization problems with interval-valued objective functions.

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 \({\mathcal{W}}\)-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.

Optimality conditions for a class of E-differentiable vector optimization problems with interval-valued objective functions under E-invexity

In this paper, a class of E-differentiable multiobjective programming with multiple objective functions and with both inequality and equality constraints is considered. The so-called E-Karush–Kuhn–Tucker necessary optimality conditions for a (weak LS-Pareto) LS-Pareto solution are established for the considered E-differentiable vector optimization problem with the multiple interval-valued objective functions. Also several sufficient optimality conditions for both LU and LS order relations are derived for such interval-valued vector optimization problems under (generalized) E-invexity hypotheses.

Solving the multi-objective stochastic interval-valued linear fractional integer programming problem

In this paper, we consider a Multi-Objective Stochastic Interval-Valued Linear Fractional Integer Programming problem (MOSIVLFIP). We especially deal with a multi-objective stochastic fractional problem involving an inequality type of constraints, where all quantities on the right side are log-normal random variables, and the objective functions coefficients are fractional intervals. The proposed solving procedure is divided in three steps. In the first one, the probabilistic constraints are converted into deterministic ones by using the chance constrained programming technique. Then, the second step consists of transforming the studied problem objectives on an optimization problem with an interval-valued objective functions. Finally, by introducing the concept of weighted sum method, the equivalent converted problem obtained from the two first steps is transformed into a single objective deterministic fractional problem. The effectiveness of the proposed procedure is illustrated through a numerical example.

The KKT optimality conditions for optimization problem with interval-valued objective function on Hadamard manifolds

ABSTRACT In this paper, we study the Karush–Kuhn–Tucker optimality conditions in an optimization problem with interval-valued objective function on Hadamard manifolds. The gH-directional differentiability for interval-valued function is defined by using the generalized Hukuhara difference. The concepts of interval-valued convexity and pseudoconvexity are introduced on Hadamard manifolds, and several properties involving such functions are also given. Under these settings, we derive the KKT optimality conditions and give a numerical example to show that the results obtained in this paper are more general than the corresponding conclusions of Wu [The Karush–Kuhn–Tucker optimality conditions in an optimization problem with interval-valued objective function. Eur J Oper Res. 2007;176:46–59] in solving the optimization problem with interval-valued objective function.

Karush-Kuhn-Tucker optimality conditions for a class of robust optimization problems with an interval-valued objective function

Abstract In this article, we study the nonlinear and nonsmooth interval-valued optimization problems in the face of data uncertainty, which are called interval-valued robust optimization problems (IVROPs). We introduce the concept of nondominated solutions for the IVROP. If the interval-valued objective function f and constraint functions g i {g}_{i} are nonsmooth on Banach space E, we establish a nonsmooth and robust Karush-Kuhn-Tucker optimality theorem.

Necessary and sufficient optimality conditions for non-linear unconstrained and constrained optimization problem with interval valued objective function

In the theory of non-linear constrained optimization problem, the most familiar Karush Kuhn Tucker (KKT) conditions play an important role. Basically, these conditions are necessary optimality conditions of a constrained optimization problem with equality and inequality constraints. The goal of this paper is to introduce the necessary optimality conditions (named as equivalent KKT conditions) of a constrained optimization problem with interval-valued objective function. Depending on the type of objective function and inequality constraints, all possible cases (i.e., objective function interval-valued and constraints are real-valued, objective function and constraints both are interval-valued, the objective function is real-valued and constraints are interval-valued) have been investigated to find the necessary optimality conditions. For this purpose, this paper deals with the necessary and sufficient conditions of unconstrained optimization problem with interval-valued objective function. Also, with the help of interval order relations the definitions of local and global optima of interval-valued function have been proposed. Finally, in order to illustrate the equivalent KKT conditions of the interval-valued constrained optimization problem and also the necessary as well as sufficient conditions of an unconstrained optimization problem, some numerical examples have been considered and solved.

The $ F $-objective function method for differentiable interval-valued vector optimization problems

<p style='text-indent:20px;'>In this paper, a differentiable vector optimization problem with the multiple interval-valued objective function and with both inequality and equality constraints is considered. The Karush-Kuhn-Tucker necessary optimality conditions are established for such a differentiable interval-valued multiobjective programming problem. Further, a new approach, called <inline-formula><tex-math id="M2">\begin{document}$ F $\end{document}</tex-math></inline-formula>-objective function method, is introduced for solving the considered differentiable vector optimization problem with the multiple interval-valued objective function. In this method, its associated vector optimization problem with the multiple interval-valued <inline-formula><tex-math id="M3">\begin{document}$ F $\end{document}</tex-math></inline-formula>-objective function is constructed. Their equivalence is established under <inline-formula><tex-math id="M4">\begin{document}$ F $\end{document}</tex-math></inline-formula>-convexity assumptions. It is shown that the introduced approach can be used to solve a nonlinear nonconvex interval-valued optimization problem. By using the introduced approximation method, it is also presented in some cases that a nonlinear nonconvex interval-valued optimization problem can be solved by the help of methods for solving linear interval-valued optimization problems.