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

doi:10.1016/j.amc.2011.09.041

Applied Mathematics and Computation

Volume 218, Issue 8, 15 December 2011, Pages 4119-4127

https://doi.org/10.1016/j.amc.2011.09.041 Get rights and content

Abstract

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.

Introduction

Convexity plays an important role in many aspects of mathematical programming including optimality conditions, duality theorems, and alternative theorems. Due to insufficiency of convexity notion in many mathematical models used in decision science, economics, engineering, and so forth, there has been an increasing interest in relaxing convexity assumptions in connection with sufficiency and duality theorems. One of the most useful generalizations of convexity is owed to Hanson [4], which was named invex by Craven [2].

Data in many real-life engineering and economical problems suffer from inexactness. Consequently, interval programming is one way to tackle these situations because it does not require the specification or the assumption of probabilistic distributions (as in stochastic programming) or possibilistic distributions (as in fuzzy programming). The modelling of the data imprecision can be made also by a fusion of fuzziness and randomness rather than either fuzziness or randomness. Simultaneous considerations of fuzziness and randomness lead us to mathematical programming under fuzzy stochastic environments. Ishibuchi and Tanaka [6] focus his study on a mathematical programming problem whose objective function has interval coefficients and proposed the ordering relation between two closed intervals by considering the maximization and minimization problems separately. Chanas and Kuchta [1] presented an approach to unify the solution methods proposed by Ishibuchi and Tanaka [6].

The interval-valued optimization problems are closely related with the inexact linear programming problems. Steuer [16] proposed three algorithms, called the F-cone algorithm, E-cone algorithm and all emanating algorithms to solve the linear programming problems with interval objective functions.

Mráz [8] provided algorithms to compute the exact upper and lower bounds of optimal solutions for linear programming problems with interval coefficients. Sengupta and Pal [11] presented an algorithm for the shortest path problem when the connected arcs in a transportation network are represented as interval numbers. The algorithm considered fuzzy preference ordering of intervals [10] from pessimistic and optimistic decision maker’s point of view. In [7], a possibilistic portfolio selection model with interval centre values is presented based on the modality approach in possibilistic programming. Urli and Nadeau [17] used an interactive method to solve the multiobjective linear programming problems with interval coefficients. Portfolio selection problem with interval objective functions was investigated by Ida [5].

Wu [18], [20] derived Karush–Kuhn–Tucker conditions for multiobjective programming problems with interval-valued objective functions and the concepts of Pareto optimal solutions are proposed by considering two orderings on the class of all closed intervals. Recently, Wu [19] discussed the Wolfe’s duality theorems for an interval-valued programming problem. Oliveria and Antunes [9] provided an overview of multiobjective linear programming problems with interval coefficients by illustrating many numerical examples.

In [13], Stancu-Minasian and Tigan presented two approaches for the inexact multiobjective programming with inexactness in the objective functions, which correspond to the conservative [12] and non-conservative [3] ways for mathematical programming with set coefficients.

In [15], Stancu-Minasian and Tigan presented some approaches for multiobjective fractional programming with set coefficients in the objective functions. Among other cases, they discuss for linear fractional multiobjective programming some conservative and nonconservative approaches. The vectorial max–min optimization is also used as a conservative approach for inexact multiobjective programming, see also [14] for the same problems where moreover a stochastic bilinear-fractional max–min problem with separate linear constraint is considered.

In this paper, we consider a class of interval-valued programming problems involving generalized convexity. By utilizing the concept of generalized convexity, we obtain sufficient optimality conditions and Mond–Weir and Wolfe type duality relations for interval-valued programming problems.

This paper is organized as follows. Some definitions, notation and some basic arithmetic of intervals are given in Section 2. In Section 3, we derive sufficient optimality conditions for a class of interval-valued programming problems under the assumption of generalized convexity. We discuss duality between the primal problem and its two types of dual models in Sections 4 Mond–Weir type duality, 5 Wolfe type duality, respectively. Conclusions and further developments are presented in Section 6.

Section snippets

Notation and preliminaries

Throughout this paper, let Rⁿ be the n-dimensional Euclidean space and $R_{+}^{n}$ be its nonnegative orthant, and let X be a nonempty open subset of Rⁿ. First we recall the following definitions.

Definition 2.1

A differentiable function f : X → R is (strict) invex with respect to η : X × X → Rⁿ at a ∈ X if for all x ∈ Rⁿ, $f (x) - f (a) (>) ⩾ η (x, a)^{T} \nabla f (a) .$

Definition 2.2

A differentiable function f : X → R is pseudo-invex with respect to η : X × X → Rⁿ at a ∈ X if for all x ∈ Rⁿ, $η (x, a)^{T} \nabla f (a) ⩾ 0 \Rightarrow f (x) - f (a) ⩾ 0,$ equivalently, $f (x) - f (a) < 0 \Rightarrow η (x, a)^{T} \nabla f (a) < 0 .$

Definition 2.3

A differentiable

Sufficient optimality conditions

In this section, we establish a Karush–Kuhn–Tucker-type sufficient optimality conditions under generalized convex functions.

Theorem 3.1 Sufficiency

Let $\bar{x} \in X_{0}$ be a feasible solution of (IPP). Assume that there exist $ξ = [ξ^{L}, ξ^{U}] > 0$ and ${\bar{λ}}_{i} ⩾ 0, i \in I (\bar{x})$ , such that

(i)
$ξ^{L} \nabla F^{L} (\bar{x}) + ξ^{U} \nabla F^{U} (\bar{x}) + \sum_{i \in I (\bar{x})} {\bar{λ}}_{i} \nabla g_{i} (\bar{x}) = 0,$
(ii)
$(ξ^{L} F^{L} + ξ^{U} F^{U}) is pseudo - invex at \bar{x} \in X_{0} and \sum_{i \in I (\bar{x})} {\bar{λ}}_{i} g_{i} (\cdot) is quasi - invex at \bar{x} \in X_{0} .$

Then

\bar{x}

is an efficient solution of (IPP).

Proof

Suppose contrary to the result that $\bar{x}$ is not an efficient solution of (IPP). Then there exists a feasible solution x

Mond–Weir type duality

In this section, we consider the following Mond–Weir type dual problem for (IPP) and present weak, strong and strict converse duality theorems. $\begin{matrix} (IPMWD) Min F (y), \\ subject to ξ^{L} \nabla F^{L} (y) + ξ^{U} \nabla F^{U} (y) + \sum_{i = 1}^{m} λ_{i} \nabla g_{i} (y) = 0, \end{matrix}$ $\sum_{i = 1}^{m} λ_{i} g_{i} (y) ⩾ 0,$ $ξ = [ξ^{L}, ξ^{U}] ⩾ 0, ξ \neq 0, λ = (λ_{1}, λ_{2}, \dots, λ_{m}) ⩾ 0,$ $y \in Y,$ where Y is an open subset of Rⁿ. $F (y) = [F^{L} (y), F^{U} (y)]$ is an interval-valued function.

Theorem 4.1 Weak duality

Let x and (y, ξ, λ) be feasible solutions for (IPP) and (IPMWD), respectively. Let $ξ = [ξ^{L}, ξ^{U}] > 0$ and assume that $(ξ^{L} F^{L} + ξ^{U} F^{U})$ be pseudo-invex at y and $\sum_{i = 1}^{m} λ_{i} g_{i} (\cdot)$ is

Wolfe type duality

In this section, we consider the following Wolfe type dual problem: $\begin{matrix} (IPWD) Min F (y) + \sum_{i = 1}^{m} λ_{i} g_{i} (y) e, \\ subject to ξ^{L} \nabla F^{L} (y) + ξ^{U} \nabla F^{U} (y) + \sum_{i = 1}^{m} λ_{i} \nabla g_{i} (y) = 0, \end{matrix}$ $ξ = [ξ^{L}, ξ^{U}] ⩾ 0, ξ \neq 0, λ = (λ_{1}, λ_{2}, \dots, λ_{m}) ⩾ 0,$ $y \in Y,$ where Y is an open subset of Rⁿ. $F (y) = [F^{L} (y), F^{U} (y)]$ is an interval-valued vector function and e = (1, 1, … , 1) such that $ξ e = [ξ^{L} e, ξ^{U} e] = [1, 1] = 1$ .

Theorem 5.1 Weak duality

Let x and (y, ξ, λ) be feasible solutions for (IPP) and (IPWD), respectively. Let $ξ = (ξ^{L}, ξ^{U}) > 0$ and assume that $(ξ^{L} F^{L} + ξ^{U} F^{U})$ be invex at y and $\sum_{i = 1}^{m} λ_{i} g_{i} (\cdot)$ is invex at y.

Then the following cannot

Conclusion

In this paper, we were concerned with a class of interval-valued programming problems and sufficient optimality conditions are discussed. The two types of dual models are considered for interval-valued programming problem. We have established weak, strong and strict converse duality theorems in the framework of generalized convexity in order to relate the efficient solutions of primal and dual problems. Sufficient optimality conditions and duality results for a class of interval-valued

Acknowledgements

The authors are thankful to the anonymous referees for their valuable suggestions, which have substantially improved the presentation of the paper.

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¹: Partially supported by the Indian School of Mines, Dhanbad, India under FRS (17)/2010-2011/AM.

²: Permanent address: Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India.

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

Abstract

Introduction

Section snippets

Notation and preliminaries

Sufficient optimality conditions

Mond–Weir type duality

Wolfe type duality

Conclusion

Acknowledgements

Eur. J. Oper. Res.

J. Math. Anal. Appl.

Appl. Math. Lett.

Eur. J. Oper. Res.

Eur. J. Oper. Res.

Eur. J. Oper. Res.

Eur. J. Oper. Res.

J. Math. Anal. Appl.

Eur. J. Oper. Res.

Invex functions and constrained local minima

Bull. Austral. Math. Soc.