Interval-valued Objective Function Research Articles

Karush–Kuhn–Tucker optimality conditions and duality for convex semi-infinite programming with multiple interval-valued objective functions

This paper deals with convex semi-infinite programming with multiple interval-valued objective functions. We first investigate necessary and sufficient Karush–Kuhn–Tucker optimality conditions for some types of optimal solutions. Then, we formulate types of Mond–Weir and Wolfe dual problems and explore duality relations under convexity assumptions. Some examples are provided to illustrate the advantages of our results in some cases.

Analyzing interval and multi-choice bi-level programming for Stackelberg game using intuitionistic fuzzy programming

The aim of this paper is to provide a computational algorithm for solving interval-valued and multi-choice bi-level programming for Stackelberg game using intuitionistic fuzzy approach. It also includes the cost parameters of upper and lower-level objective functions corresponding to interval numbers and multi-choice types while the parameters of constraints are intuitionistic fuzzy numbers. The interval-valued and multi-choice types objective functions are converted into deterministic form using interval programming and a general transformation technique respectively. Again, the intuitionistic fuzzy parameters of constraints are reduced to an interval by taking expected value of intuitionistic fuzzy number. A conflicting nature between the objective functions is resolved with the help of intuitionistic fuzzy programming by considering nonlinear degree of membership and non-membership functions respectively. The resultant max-min problem is solved with LINGO 15.0 iterative scheme. The developed algorithm is illustrated by an application example and Pareto optimality test is performed as well. The conclusions and an outlook of future study are described at last.

Karush-Kuhn-Tucker optimality conditions and duality for a semi-infinite programming with multiple interval-valued objective functions

Karush-Kuhn-Tucker optimality conditions and duality for a semi-infinite programming with multiple interval-valued objective functions

Exactness Property of the Exact Absolute Value Penalty Function Method for Solving Convex Nondifferentiable Interval-Valued Optimization Problems

In the paper, the classical exact absolute value function method is used for solving a nondifferentiable constrained interval-valued optimization problem with both inequality and equality constraints. The property of exactness of the penalization for the exact absolute value penalty function method is analyzed under assumption that the functions constituting the considered nondifferentiable constrained optimization problem with the interval-valued objective function are convex. The conditions guaranteeing the equivalence of the sets of LU-optimal solutions for the original constrained interval-valued extremum problem and for its associated penalized optimization problem with the interval-valued exact absolute value penalty function are given.

Optimality conditions and duality results for nonsmooth vector optimization problems with the multiple interval-valued objective function

In this paper, both Fritz John and Karush-Kuhn-Tucker necessary optimality conditions are established for a (weakly) LU-efficient solution in the considered nonsmooth multiobjective programming problem with the multiple interval-objective function. Further, the sufficient optimality conditions for a (weakly) LU-efficient solution and several duality results in Mond-Weir sense are proved under assumptions that the functions constituting the considered nondifferentiable multiobjective programming problem with the multiple interval-objective function are convex.

An approach for solving a fuzzy bilevel programming problem through nearest interval approximation approach and KKT optimality conditions

In this paper, we consider a kind of bilevel linear programming problem where the coefficients of both objective functions are fuzzy numbers. In order to deal with such a problem, the original problem can be approximated by an interval bilevel programming problem in terms of the nearest interval approximation of a fuzzy number. Based on the Karush–Kuhn–Tucker (KKT) optimality conditions for the optimization problem with an interval-valued objective function, the interval bilevel programming problem can be converted into a single-level programming problem with an interval-value objective function. To minimize the interval objective function, the order relations of interval numbers are used to transform the uncertain single-objective optimization into a multi-objective optimization solved by global criteria method (GCM). Finally, illustrative numerical examples are provided to demonstrate the feasibility of the proposed approach.

Reply to a Note on the Paper “Duality Theory for Optimization Problems with Interval-Valued Objective Function”

The author replies to a note on the paper "Duality Theory for Optimization Problems with Interval-Valued Objective Function" by saying that the note is invalid.

Newton method to obtain efficient solutions of the optimization problems with interval-valued objective functions

In this paper, a Newton method is proposed to obtain efficient solutions for the optimization problems with interval-valued objective functions. In the concept of efficient solution of the problem, a suitable partial ordering for a pair of intervals is used. Through the notion of generalized Hukuhara difference of a pair of intervals, the generalized Hukuhara differentiability of multi-variable interval-valued functions is defined and analyzed to develop the proposed method. The objective function in the problem is assumed to be twice continuously generalized Hukuhara differentiable. Under this hypothesis, it is exhibited that the method has a local quadratic rate of convergence. In order to improve the local convergence of the method to a global convergence, an updated Newton method is also given. The sequential algorithms and the convergence results of the proposed methods are demonstrated. Several numerical examples are presented to illustrate the proposed methodologies.

A multiobjective interval goal programming method using penalty function

This article demonstrates how the penalty function can efficiently be used for solving multiobjective linear programming problem with interval valued objective functions. In the proposed approach, first the target intervals are obtained for the defined objectives. Then, using interval arithmetic and the concept of interval goal programming (IGP) the interval goals are transformed into crisp goals. In the process of solution, the goal achievement function is constructed as a convex combination of the sum of total penalty and the maximum penalty associated with unwanted deviations of the respective goals. Numerical examples are provided to illustrate the proposed approach.

A multiobjective interval goal programming method using penalty function

This article demonstrates how the penalty function can efficiently be used for solving multiobjective linear programming problem with interval valued objective functions. In the proposed approach, first the target intervals are obtained for the defined objectives. Then, using interval arithmetic and the concept of interval goal programming (IGP) the interval goals are transformed into crisp goals. In the process of solution, the goal achievement function is constructed as a convex combination of the sum of total penalty and the maximum penalty associated with unwanted deviations of the respective goals. Numerical examples are provided to illustrate the proposed approach.

A Note on the Paper “Duality Theory for Optimization Problems with Interval-Valued Objective Functions”

With this note, we want to open a discussion on the duality theory for Interval Optimization. We begin by showing that some duality results Wu ( J Optim Theory Appl 144:615---628, 2010) are useless.

On interval-valued invex mappings and optimality conditions for interval-valued optimization problems

In this paper, we first introduce the concept of interval-valued invex mappings by using gH-differentiability and compare it with interval-valued weakly invex mappings. We can observe that interval-valued invex mappings are more general than interval-valued weakly invex mappings. In addition, the sufficient optimality condition for interval-valued objective functions is derived under invexity.

A new multi-section based technique for constrained optimization problems with interval-valued objective function

In this paper, an efficient optimization technique is proposed for constrained optimization problems with interval valued objective function. At first, the significance of interval-valued objective function and the meaning of the interval-valued solution of the proposed problem have been explained with graphical interpretation. Generally, this type of problems has infinitely many compromise solutions. The aim of this approach is to obtain one of such solutions with higher accuracy and lower computational cost. The proposed technique is mainly based on the splitting criterion of the accepted/prescribed search region, calculation of the interval inclusion functions and the selection of subregion depending on the modified interval order relations in the context of the decision makers’ point of view. Novel interval oriented constraint satisfaction rules are used for non-interval equality and inequality constraints. Clearly, the proposed technique is nothing but an imitation of well known interval analysis based branch and bound (B &B) optimization approach. The modified multi-section division criterion with some new interval oriented constraint satisfaction rules for non-interval-valued equality and inequality constraints and some novel interval order relations in the context of the decision makers’ point of view have been applied to increase the efficiency of the proposed algorithm. Finally, the technique is applied for solving some test problems and the results are compared with the same obtained from the existing methods.

Duality and saddle-point type optimality for interval-valued programming

In this paper, Mond-Weir’s type dual in programming problem with an interval-valued objective function and interval-valued inequality constrict conditions is formulated. Duality theorems are established under suitable conditions. A real-valued Lagrangian function for the interval-valued programming is defined. Further, the saddle point of Lagrangian function is also defined and saddle point optimality conditions are presented.

Optimality conditions of type KKT for optimization problem with interval-valued objective function via generalized derivative

This paper addresses the optimization problems with interval-valued objective function. For this we consider two types of order relation on the interval space. For each order relation, we obtain KKT conditions using of the concept of generalized Hukuhara derivative ( $$gH$$ -derivative) for interval-valued functions. The $$gH$$ -derivative is a concept more general of derivative for this class of functions than other concepts of derivative. We make some comparison with previous result given by other authors and we show some advantages of our result.

The KKT optimality conditions in a class of generalized convex optimization problems with an interval-valued objective function

In this paper, we study the Karush–Kuhn–Tucker optimality conditions in a class of nonconvex optimization problems with an interval-valued objective function. Firstly, the concepts of preinvexity and invexity are extended to interval-valued functions. Secondly, several properties of interval-valued preinvex and invex functions are investigated. Thirdly, the KKT optimality conditions are derived for LU-preinvex and invex optimization problems with an interval-valued objective function under the conditions of weakly continuous differentiablity and Hukuhara differentiablity. Finally, the relationships between a class of variational-like inequalities and the interval-valued optimization problems are established.

Solving the Interval-Valued Linear Fractional Programming Problem

This paper introduces an interval valued linear fractional programming problem (IVLFP). An IVLFP is a linear fractional programming problem with interval coefficients in the objective function. It is proved that we can convert an IVLFP to an optimization problem with interval valued objective function which its bounds are linear fractional functions. Also there is a discussion for the solutions of this kind of optimization problem.

THE KARUSH-KUHN-TUCKER OPTIMALITY CONDITIONS IN INTERVAL-VALUED MULTIOBJECTIVE PROGRAMMING PROBLEMS

The Karush-Kuhn-Tucker (KKT) necessary optimality conditions for nonlinear differentiable programming problems are also sufficient under suitable convexity assumptions. The KKT conditions in multiobjective programming problems with interval-valued objective and constraint functions are derived in this paper. The main contribution of this paper is to obtain the Pareto optimal solutions by resorting to the sufficient optimality condition.

Duality Theory for Optimization Problems with Interval-Valued Objective Functions

A solution concept in optimization problems with interval-valued objective functions, which is essentially similar to the concept of nondominated solution in vector optimization problems, is introduced by imposing a partial ordering on the set of all closed intervals. The interval-valued Lagrangian function and interval-valued Lagrangian dual function are also proposed to formulate the dual problem of the interval-valued optimization problem. Under this setting, weak and strong duality theorems can be obtained.

The Karush–Kuhn–Tucker optimality conditions in multiobjective programming problems with interval-valued objective functions

The KKT conditions in multiobjective programming problems with interval-valued objective functions are derived in this paper. Many concepts of Pareto optimal solutions are proposed by considering two orderings on the class of all closed intervals. In order to consider the differentiation of an interval-valued function, we invoke the Hausdorff metric to define the distance between two closed intervals and the Hukuhara difference to define the difference of two closed intervals. Under these settings, we are able to consider the continuity and differentiability of an interval-valued function. The KKT optimality conditions can then be naturally elicited.