Generalized Invexity and Duality in Multiobjective Programming Problems

In this paper, we are concerned with the multiobjective programming problem with inequality constraints. We introduce new classes of generalized type I vector-valued functions. Duality theorems are proved for Mond–Weir and general Mond–Weir type duality under the above generalized type I assumptions.     

Perturbation Approach to Sensitivity Analysis in Mathematical Programming

This paper presents a perturbation approach for performing sensitivity analysis of mathematical programming problems. Contrary to standard methods, the active constraints are not assumed to remain active if the problem data are perturbed, nor the partial derivatives are assumed to exist. In other words, all the elements, variables, parameters, Karush–Kuhn–Tucker multipliers, and objective function values may vary provided that optimality is maintained and the general structure of a feasible perturbation (which is a polyhedral cone) is obtained. This allows determining: (a) the local sensitivities, (b) whether or not partial derivatives exist, and (c) if the directional derivative for a given direction exists. A method for the simultaneous obtention of the sensitivities of the objective function optimal value and the primal and dual variable values with respect to data is given. Three examples illustrate the concepts presented and the proposed methodology. Finally, some relevant conclusions are drawn. The authors are indebted to the Ministry of Science and Education of Spain, Projects CICYT DPI2002-04172-C04-02 and CICYT DPI2003-01362, and to the Fulbright Commission for partial support. The authors are grateful to the referees for comments improving the quality of the paper.     

Generalized concavity and duality with a square root term

Duality results for a class of nondifferentiable mathematical programming problems are given. These results allow for the weakening of the usual convexity conditions required for duality to hold. A pair of symmetric and self dual nondifferentiable programs under weaker convexity conditions are also given. A subgradient symmetric duality is proposed and its limitations discussed. Finally, a pair of nondifferentiable mathematical programs containing arbitrary norms is presented.     

(p,r)-不变凸函数规划问题的鞍点定理

本文首先介绍了一个广义Lagrange向量函数L(x,u),并利用一类新的广义 凸函数:(p,r)——不变凸函数讨论了多目标分式规划问题的鞍点最优性条件．     

Duality in Fuzzy Linear Programming: Some New Concepts and Results

A class of fuzzy linear programming (FLP) problems based on fuzzy relations is introduced, the concepts of feasible and -efficient solutions are defined. The class of crisp (classical) LP problems and interval LP problems can be embedded into the class of FLP ones. Moreover, for FLP problems a new concept of duality is introduced and the weak and strong duality theorems are derived. The previous results are applied to the special case of interval LP and compared to the existing literature.     

Higher-Order Duality in Nondifferentiable Minimax Programming with Generalized Type I Functions

A unified higher-order dual for a nondifferentiable minimax programming problem is formulated. Weak, strong and strict converse duality theorems are discussed involving generalized higher-order (F,α,ρ,d)-Type I functions. The research of second author was supported by the Department of Atomic Energy, Government of India, under the NBHM Post Doctoral Fellowship Program 40/9/2005-R&D II/2398.     

On semi-infinite minmax programming with generalized invexity

Here, we consider the minmax programming problem with a set of restrictions indexed in a compact. As a novelty, we obtain optimality criteria of the Kuhn--Tucker type involving a limited number of restrictions and prove both necessity and sufficiency under new weaker invexity assumptions. Also some dual problems are introduced and it is proved that the weak and strong duality properties hold within the same environment.     

Mathematical Programming Techniques in Constraint Programming: A Short Overview

In recent years, the integration of techniques from Artificial Intelligence and Operations Research has shown to improve the solutions of complex and large scale combinatorial optimization problems, in terms of efficiency, scalability and optimality. In this context, Constraint Programming is an emerging discipline situated at the confluence of the two fields that has been recognized as a suitable environment for achieving such an integration. This paper briefly presents the integration directions explored in the literature, and provides some pointers to relevant work in these directions.     

Parameter-Free Dual Models for Fractional Programming with Generalized Invexity

By parameter-free approach, we establish sufficient optimality conditions for nondifferentiable fractional variational programming under certain specific structure of generalized invexity. Employing the sufficient optimality conditions, two parameter-free dual models are formulated. The weak duality, strong duality and strict converse duality theorems are proved in the framework of generalized invexity.     

Generalized Invariant Monotonicity and Invexity of Non-differentiable Functions

This paper is devoted to the study of relationships between several kinds of generalized invexity of locally Lipschitz functions and generalized monotonicity of corresponding Clarke’s subdifferentials. In particular, some necessary and sufficient conditions of being a locally Lipschitz function invex, quasiinvex or pseudoinvex are given in terms of momotonicity, quasimonotonicity and pseudomonotonicity of its Clarke’s subdifferential, respectively. As an application of our results, the existence of the solutions of the variational-like inequality problems as well as the mathematical programming problems (MP) is given. Our results extend and unify the well known earlier works of many authors.     

具广义V-不变凸多目标变分的混合对偶性

在函数广义V-不变凸性的条件下,建立了多目标变分关于有效解的混合对偶理论.     

集函数多目标分式规划的最优性和广义凸对偶性

本文讨论了集函数多目标（分母不同）分式规划，给出了Ｇｅｏｆｆｒｉｏｎ正常有效解的必要和充分条件，并讨论了关于有效解的广义凸对偶理论．     

Generalized Nonlinear Lagrangian Formulation for Bounded Integer Programming

Several nonlinear Lagrangian formulations have been recently proposed for bounded integer programming problems. While possessing an asymptotic strong duality property, these formulations offer a success guarantee for the identification of an optimal primal solution via a dual search. Investigating common features of nonlinear Lagrangian formulations in constructing a nonlinear support for nonconvex piecewise constant perturbation function, this paper proposes a generalized nonlinear Lagrangian formulation of which many existing nonlinear Lagrangian formulations become special cases.     

一类数学规划问题的公式解

本文运用矩阵的知识,讨论一类数学规划问题的公式解。     

Banach空间中对偶规划的存在性定理

本文利用无穷下降方向，在Ｂａｎａｃｈ空间中建立了若干非凸对偶规划的存在性定理。作为特例，给出了半无限对偶规划和有限对偶规划的存在性定理。     

Optimality Conditions and Duality in Nondifferentiable Minimax Fractional Programming with Generalized Convexity

We establish sufficient optimality conditions for a class of nondifferentiable minimax fractional programming problems involving (F, α, ρ, d)-convexity. Subsequently, we apply the optimality conditions to formulate two types of dual problems and prove appropriate duality theorems. The authors thank the referee for valuable suggestions improving the presentation of the paper.     

Optimally Conditions for Pareto Nonsmooth Nonconvex Programming in Banach Spaces

In this paper, we consider a vector optimization problem where all functions involved are defined on Banach spaces. We obtain necessary and sufficient criteria for optimality in the form of Karush–Kuhn–Tucker conditions. We also introduce a nonsmooth dual problem and provide duality theorems.     

Duality Theorems of Multiobjective Programming for a Class of Generalized Convex Functions

1.DefinitionsDefinition1.AfunctionalF(x)inthespaceVCE"issaidtobeasublinearfunctionalifforx,yeV,andor20,Inparticular,F(0)=0.Letop(x)beadifferentiablerealfunctiononasetCCEd.ForagivensublinearfunctionFandafunctionp:CxC-EIIp(x,u)/0(x/u),themoregeneralgeneralizedconvexfunctioncanbedefinedasthefollwing:Definition2.op(x)issaidtobe(F,p)--invarialltconvexfunctiononCifforxl,xZECDefinition3.op(x)issaidtobe(F,P)--invariantquasiconvexfunctiononCifforal,xZECthatis,Definition4.op(x)issaidtobe(F,…     

数学规划的求解与广义Hopfield神经网络

神经网络技术最为成功的应用领域之一是用于求解优化问题,本文就近年来的求解优化问题的神经网络方法进行了综述     

一类或决策模型及其应用

提出了一类特殊类型的数学规划模型并给出了一种新的分枝定界算法.这类数学模型尽管可以转化为0-1规划模型,但它相对于转化后的0-1规划模型:①决策意义明确,表达形式相对简单;②不需要引入参数M并在求解前确定其上界;③相对于求解转化后的0-1规划模型的分枝定界法,新分枝定界算法在最好情形下计算量最多为原算法的八分之一.作为本模型的一个应用,可以用来解决一些要么不实施要么有一定数量下限限制才可以实施的决策问题.