Symmetric duality for minimax mixed integer programming problems with pseudo-invexity

In this paper, we consider the pair of symmetric dual multiobjective variational mixed integer programs proposed by Chen and Yang [X. Chen, J. Yang, Symmetric duality for minimax multiobjective variational mixed integer programming problems with partial-invexity, European Journal of Operational Research 181 (2007) 76-85.] and extend some of their results under the assumptions of partial-pseudo-invexity and separability on the functions involved. These results include several results available in literature as special cases.     

Symmetric duality for a class of nondifferentiable multi-objective fractional variational problems

We introduce a symmetric dual pair for a class of nondifferentiable multi-objective fractional variational problems. Weak, strong, converse and self duality relations are established under certain invexity assumptions. The paper includes extensions of previous symmetric duality results for multi-objective fractional variational problems obtained by Kim, Lee and Schaible [D.S. Kim, W.J. Lee, S. Schaible, Symmetric duality for invex multiobjective fractional variational problems, J. Math. Anal. Appl. 289 (2004) 505-521] and symmetric duality results for the static case obtained by Yang, Wang and Deng [X.M. Yang, S.Y. Wang, X.T. Deng, Symmetric duality for a class of multiobjective fractional programming problems, J. Math. Anal. Appl. 274 (2002) 279-295] to the dynamic case.     

Symmetric duality for invex multiobjective fractional variational problems

We introduce a symmetric dual for multiobjective fractional variational problems. Under certain invexity assumptions we establish weak, strong and converse duality theorems as well as self-duality relations. The paper includes an extension of previous symmetric duality results for the static case obtained by Weir to the dynamic case.     

Mixed symmetric duality in non-differentiable multiobjective mathematical programming

Two mixed symmetric dual models for a class of non-differentiable multiobjective nonlinear programming problems with multiple arguments are introduced in this paper. These two mixed symmetric dual models unify the four existing multiobjective symmetric dual models in the literature. Weak and strong duality theorems are established for these models under some mild assumptions of generalized convexity. Several special cases are also obtained.     

含有锥约束的多目标变分问题的广义对称对偶性

本文研究了一类含有锥约束多目标变分问题的广义对称对偶性.利用函数的(F,ρ)-不变凸性的条件,得出了多目标变分问题关于有效解的弱对偶定理、强对偶定理和逆对偶定理,将多目标变分问题的对称对偶性理论推广到含有锥约束的广义对称对偶性上来.     

多目标分式变分问题的混合对偶性

在广义B-Ⅰ凸性条件下,建立了多目标分式变分问题的混合对偶模型,使得M ond-W e ir型对偶和W o lfe型成为其特殊情况,并建立了关于有效解的混合对偶理论.     

Symmetric duality for minimax multiobjective variational mixed integer programming problems with partial-invexity

A pair of symmetric dual multiobjective variational mixed integer programs for the polars of arbitrary cones are formulated, which some primal and dual variables are constrained to belong to the set of integers. Under the separability with respect to integer variables and partial-invexity assumptions on the functions involved, we prove the weak, strong, converse and self-duality theorems to related minimax efficient solution. These results include some of available results.     

具B-I凸性条件下变分控制问题的混合对偶性

在广义B-凸性条件下,建立了多目标变分控制问题的混合对偶模型,使得Mond-Weir型和Wolfe型对偶成为其特殊情况,并建立了关于有效解的混合对偶理论.     

On a pair of nonlinear mixed integer programming problems

We consider maximin and minimax nonlinear mixed integer programming problems which are nonsymmetric in duality sense. Under weaker (pseudo-convex/pseudo-concave) assumptions, we show that the supremum infimum of the maximin problem is greater than or equal to the infimum supremum of the minimax problem. As a particular case, this result reduces to the weak duality theorem for minimax and symmetric dual nonlinear mixed integer programming problems. Further, this is used to generalize available results on minimax and symmetric duality in nonlinear mixed integer programming.     

Second order symmetric duality for nonlinear multiobjective mixed integer programming

We formulate two pairs of second order symmetric duality for nonlinear multiobjective mixed integer programs for arbitrary cones. By using the concepts of efficiency and second order invexity, we establish weak, strong, converse and self-duality theorems for the dual models. Several known results are obtained as special cases.     

Higher-order nondifferentiable symmetric duality with generalized F-convexity

A pair of nondifferentiable higher-order Wolfe type symmetric dual models is formulated and usual duality theorems are established under higher-order F-convexity assumption. Symmetric minimax mixed integer primal and dual problems are also discussed.     

多目标变分问题的混合对偶性

本文给出了一类多目标变分问题的混合对偶 ,使得 Wolfe型对偶和 Mond-Weir型对偶是其特殊情况 ,并在函数 (F ,ρ) -凸性的条件下建立了多目标变分问题关于有效解的混合对偶理论 .     

Non-differentiable higher-order symmetric duality in mathematical programming with generalized invexity

A pair of non-differentiable higher-order symmetric dual model in mathematical programming is formulated. The weak and strong duality theorems are established under higher-order-invexity assumption. Symmetric minimax mixed integer primal and dual problems are also investigated.     

Symmetric duality for second-order fractional programs

In this paper, a pair of symmetric dual second-order fractional programming problems is formulated and appropriate duality theorems are established. These results are then used to discuss the minimax mixed integer symmetric dual fractional programs.     

Non-differentiable symmetric duality for multiobjective programming with cone constraints

Two pairs of non-differentiable multiobjective symmetric dual problems with cone constraints over arbitrary cones, which are Wolfe type and Mond–Weir type, are considered. On the basis of weak efficiency with respect to a convex cone, we obtain symmetric duality results for the two pairs of problems under cone-invexity and cone-pseudoinvexity assumptions on the involved functions. Our results extend the results in Khurana [S. Khurana, Symmetric duality in multiobjective programming involving generalized cone-invex functions, European Journal of Operational Research 165 (2005) 592–597] to the non-differentiable multiobjective symmetric dual problem.     

锥约束多目标规划的二阶对称对偶性质

给出一对锥约束多目标非线性规划的二阶对称对偶问题,以及二阶F凸函数类的概念.在二阶F凸假设下证明了真有效解的对偶性质———弱对偶性、强对偶性及逆对偶性.     

On efficiency and mixed duality for a new class of nonconvex multiobjective variational control problems

In this paper, we extend the notions of \((\Phi ,\rho )\) -invexity and generalized \((\Phi ,\rho )\) -invexity to the continuous case and we use these concepts to establish sufficient optimality conditions for the considered class of nonconvex multiobjective variational control problems. Further, multiobjective variational control mixed dual problem is given for the considered multiobjective variational control problem and several mixed duality results are established under \((\Phi ,\rho )\) -invexity.     

多目标分数变分问题的对偶性

本文利用参数逼近在函数广义（F，ρ）－凸的条件下，建立了一类多目标式变分问题关于有效解的对偶理论。     

On symmetric duality in nondifferentiable mathematical programming with F-convexity

Usual symmetric duality results are proved for Wolfe and Mond-Weir type nondifferentiable nonlinear symmetric dual programs under F-convexity F-concavity and F-pseudoconvexity F-pseudoconcavity assumptions. These duality results are then used to formulate Wolfe and Mond-Weir type nondifferentiable minimax mixed integer dual programs and symmetric duality theorems are established. Moreover, nondifferentiable fractional symmetric dual programs are studied by using the above programs.