Interval-valued Programming Problem Research Articles

Optimality Conditions for E-Convex Interval-Valued Programming Problem Using gH-Symmetrical Derivative

Taking into consideration gH-symmetrical derivative, we discuss Fritz John type optimality conditions for E-convex functions in interval-valued optimization problems. By a suitable example, we show that gH-symmetrical derivative is an extension of gH-derivative. We use LU type of order relation in this context.

Necessary optimality conditions for a multiobjective semi-infinite interval-valued programming problem

Necessary optimality conditions for a multiobjective semi-infinite interval-valued programming problem

On semidifferentiable interval-valued programming problems

In this paper, we consider the semidifferentiable case of an interval-valued minimization problem and establish sufficient optimality conditions and Wolfe type as well as Mond–Weir type duality theorems under semilocal E-preinvex functions. Furthermore, we present saddle-point optimality criteria to relate an optimal solution of the semidifferentiable interval-valued programming problem and a saddle point of the Lagrangian function.

Interval-Valued Programming Problem with Infinite Constraints

In this paper, we explore a class of interval-valued programming problem where constraints are interval-valued and infinite. Necessary optimality conditions are derived. Notion of generalized $$(\varPhi , \rho )-$$ invexity is utilized to establish sufficient optimality conditions. Further, two duals, namely Wolfe and Mond–Weir, are proposed for which duality results are proved.

Multi-objective immune genetic algorithm solving nonlinear interval-valued programming

This work studies one multi-objective immune genetic algorithm with small population to solve a general kind of unconstrained multi-objective interval-valued programming. In this optimization approach, those competitive individuals are discriminated based on interval arithmetic rules and a possibility model; a crowding degree model in interval-valued environments is developed to eliminate redundant individuals; the current population promotes different individuals to evolve towards specific directions by population sorting and immune evolution, while those elitist individuals found accelerate to explore the desired regions through genetic evolution. The theoretical analysis has showed that the computational complexity of the proposed approach depends mainly on the elitist population size. Comparative experiments have illustrated that the approach can take a rational tradeoff between effect and efficiency. It can perform well over the compared approaches as a whole, and has the potential to solving multi-modal and hard multi-objective interval-valued programming problems.

On optimality conditions and duality for non-differentiable interval-valued programming problems with the generalized (<i>F</i>, <i>ρ</i>)-convexity

<p style='text-indent:20px;'>Because interval-valued programming problem is used to tackle interval uncertainty that appears in many mathematical or computer models of some deterministic real-world phenomena, this paper considers a non-differentiable interval-valued optimization problem in which objective and all constraint functions are interval-valued functions, and the involved endpoint functions in interval-valued functions are locally Lipschitz and Clarke sub-differentiable. A necessary optimality condition is first established. Some sufficient optimality conditions of the considered problem are derived for a feasible solution to be an efficient solution under the $G-(F, ρ)$ convexity assumption. Weak, strong, and converse duality theorems for Wolfe and Mond-Weir type duals are also obtained in order to relate the efficient solution of primal and dual inter-valued programs.

New efficiency conditions for multiobjective interval-valued programming problems

In this paper, we focus on necessary and sufficient efficiency conditions for optimization problems with multiple objectives and a feasible set defined by interval-valued functions. A new concept of Fritz-John and Karush–Kuhn–Tucker-type points is introduced for this mathematical programming problem based on the gH-derivative concept. The innovation and importance of these concepts are presented from a practical and computational point of view. The problem is approached directly, without transforming it into a real-valued programming problem, thereby attaining theoretical results that are more powerful and computationally more efficient under weaker hypotheses. We also provide necessary conditions for efficiency, which have been inexistent in the relevant literature to date. The identification of necessary conditions is important for the development of future computational optimization techniques in an interval-valued environment. We introduce new generalized convexity notions for gH-differentiable interval-valued problems which are a generalization of previous concepts and we prove a sufficient efficiency condition based on these concepts. Finally, the efficiency conditions for deterministic programming problems are shown to be particular instances of the results proved in this paper. The theoretical developments are illustrated and justified through several numerical examples.

An Exact $$l_1$$ l 1 Penalty Approach for Interval-Valued Programming Problem

The objective of this paper is to propose an exact \(l_1\) penalty method for constrained interval-valued programming problems which transform the constrained problem into an unconstrained interval-valued penalized optimization problem. Under some suitable conditions, we establish the equivalence between an optimal solution of interval-valued primal and penalized optimization problem. Moreover, saddle-point type optimality conditions are also established in order to find the relation between an optimal solution of penalized optimization problem and saddle-point of Lagrangian function. Numerical examples are given to illustrate the derived results.

Optimality conditions for invex interval valued nonlinear programming problems involving generalized H-derivative

In this paper, some interval valued programming problems are discussed. The solution concepts are adopted from Wu [7] and Chalco-Cano et al. [34]. By considering generalized Hukuhara differentiability and generalized convexity (viz. ?-preinvexity, ?-invexity etc.) of interval valued functions, the KKT optimality conditions for obtaining (LS and LU) optimal solutions are elicited by introducing Lagrangian multipliers. Our results generalize the results of Wu [7], Zhang et al. [11] and Chalco-Cano et al. [34]. To illustrate our theorems suitable examples are also provided

Optimality conditions and duality for interval-valued optimization problems using convexifactors

This paper is devoted to the applications of convexifactors on interval-valued programming problem. Based on the concept of LU optimal solution, sufficient optimality conditions are established under generalized \(\partial ^{*}\)-convexity assumptions. Furthermore, appropriate duality theorems are derived for two types of dual problem, namely Mond–Weir and Wolfe type duals. We also construct examples to manifest the established relations.

Optimality conditions and duality in nondifferentiable interval-valued programming

In this paper, we define the concepts of $LU$ optimal solution to interval-valued programming problem. By considering the concepts of $LU$ optimal solution, the Fritz John type and Kuhn-Tucker type necessary and sufficient optimality conditions for nondifferentiable interval programming are derived. Further, we establish the dual problem and prove the duality theorems.

On sufficiency and duality for a class of interval-valued programming problems

In this paper, we are concerned with an interval-valued programming problem. Sufficient optimality conditions are established under generalized convex functions for a feasible solution to be an efficient solution. Appropriate duality theorems for Mond–Weir and Wolfe type duals are discussed in order to relate the efficient solutions of primal and dual programs.