Minimax mixed integer symmetric duality for multiobjective variational problems

A Mond–Weir type multiobjective variational mixed integer symmetric dual program over arbitrary cones is formulated. Applying the separability and generalized F-convexity on the functions involved, weak, strong and converse duality theorems are established. Self duality theorem is proved. A close relationship between these variational problems and static symmetric dual minimax mixed integer multiobjective programming problems is also presented.     

Second order symmetric duality in non-differentiable multiobjective programming with F-convexity

This paper is concerned with a pair of Mond–Weir type second order symmetric dual non-differentiable multiobjective programming problems. We establish the weak and strong duality theorems for the new pair of dual models under second order F-convexity assumptions. Several results including many recent works are obtained as special cases.     

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.     

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.     

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.     

Second order symmetric duality in multiobjective programming involving generalized cone-invex functions

In this paper, cone-second order pseudo-invex and strongly cone-second order pseudo-invex functions are defined. A pair of Mond–Weir type second order symmetric dual multiobjective programs is formulated over arbitrary cones. Weak, strong and converse duality theorems are established under aforesaid generalized invexity assumptions. A second self-duality theorem is also given by assuming the functions involved to be skew-symmetric.     

Higher-order symmetric duality in multiobjective programming problems under higher-order invexity

In this paper we present a pair of Wolfe and Mond-Weir type higher-order symmetric dual programs for multiobjective symmetric programming problems. Different types of higher-order duality results (weak, strong and converse duality) are established for the above higher-order symmetric dual programs under higher-order invexity and higher-order pseudo-invexity assumptions. Also we discuss many examples and counterexamples to justify our work.     

Optimality and Duality in Nondifferentiable and Multiobjective Programming under Generalized d-Invexity

In this paper, we are concerned with the nondifferentiable multiobjective programming problem with inequality constraints. We introduce four new classes of generalized d-type-I functions. By utilizing the new concepts, Antczak type Karush-Kuhn-Tucker sufficient optimality conditions, Mond-Weir type and general Mond-Weir type duality results are obtained for non-differentiable and multiobjective programming.     

Second-order multiobjective symmetric duality with cone constraints

In this paper, we formulate Wolfe and Mond–Weir type second-order multiobjective symmetric dual problems over arbitrary cones. Weak, strong and converse duality theorems are established under η$\eta$-bonvexity/η$\eta$-pseudobonvexity assumptions. This work also removes several omissions in definitions, models and proofs for Wolfe type problems studied in Mishra [9]. Moreover, self-duality theorems for these pairs are obtained assuming the function involved to be skew symmetric.     

Pseudoinvexity,optimality conditions and efficiency in multiobjective problems; duality

In this paper, we establish characterizations for efficient solutions to multiobjective programming problems, which generalize the characterization of established results for optimal solutions to scalar programming problems. So, we prove that in order for Kuhn–Tucker points to be efficient solutions it is necessary and sufficient that the multiobjective problem functions belong to a new class of functions, which we introduce. Similarly, we obtain characterizations for efficient solutions by using Fritz–John optimality conditions. Some examples are proposed to illustrate these classes of functions and optimality results. We study the dual problem and establish weak, strong and converse duality results.     

Symmetric duality in multiobjective programming involving generalized cone-invex functions

In this paper, cone-pseudoinvex and strongly cone-pseudoinvex functions are defined. A pair of Mond–Weir type symmetric dual multiobjective programs is formulated over arbitrary cones. Weak duality, strong duality and converse duality theorems are established using the above-defined functions. A self-duality theorem is also given by assuming the functions involved to be skew-symmetric.     

Symmetric dual multiobjective programming

In this paper a pair of symmetric dual multiobjective programming problems is formulated and the duality theorems are established for pseudo-convex/pseudo-concave functions.     

A note on higher-order nondifferentiable symmetric duality in multiobjective programming

In this work, we establish a strong duality theorem for Mond–Weir type multiobjective higher-order nondifferentiable symmetric dual programs. This fills some gaps in the work of Chen [X. Chen, Higher-order symmetric duality in nondifferentiable multiobjective programming problems, J. Math. Anal. Appl. 290 (2004) 423–435].     

Second order symmetric duality in multiobjective programming

A pair of Mond–Weir type multiobjective second order symmetric dual programs are formulated without non-negativity constraints. Weak duality, strong duality and converse duality theorems are established under η-bonvexity and η-pseudobonvexity assumptions. A second order self-duality theorem is given by assuming the functions involved to be skew-symmetric.     

A characterization of pseudoinvexity for the efficiency in non-differentiable multiobjective problems. Duality

We prove that in order for the Kuhn-Tucker or Fritz John points to be efficient solutions, it is necessary and sufficient that the non-differentiable multiobjective problem functions belong to new classes of functions that we introduce here: KT-pseudoinvex-II or FJ-pseudoinvex-II, respectively. We illustrate it by examples. These characterizations generalize recent results given for the differentiable case. We study the dual problem and establish weak, strong and converse duality results.     

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.     

Duality for multiobjective optimization problems with convex objective functions and D.C. constraints

In this paper we provide a duality theory for multiobjective optimization problems with convex objective functions and finitely many D.C. constraints. In order to do this, we study first the duality for a scalar convex optimization problem with inequality constraints defined by extended real-valued convex functions. For a family of multiobjective problems associated to the initial one we determine then, by means of the scalar duality results, their multiobjective dual problems. Finally, we consider as a special case the duality for the convex multiobjective optimization problem with convex constraints.     

Duality theorems for a new class of multitime multiobjective variational problems

In this work, we consider a new class of multitime multiobjective variational problems of minimizing a vector of functionals of curvilinear integral type. Based on the normal efficiency conditions for multitime multiobjective variational problems, we study duals of Mond-Weir type, generalized Mond-Weir-Zalmai type and under some assumptions of (??, b)-quasiinvexity, duality theorems are stated. We give weak duality theorems, proving that the value of the objective function of the primal cannot exceed the value of the dual. Moreover, we study the connection between values of the objective functions of the primal and dual programs, in direct and converse duality theorems. While the results in §1 and §2 are introductory in nature, to the best of our knowledge, the results in §3 are new and they have not been reported in literature.     

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.     

Multiobjective symmetric duality with cone constraints

We formulate a pair of multiobjective symmetric dual programs for pseudo-invex functions and arbitrary cones. Our model is unifying the Wolfe vector symmetric dual and the Mond-Weir vector symmetric dual models. We establish the weak, strong, converse and self duality theorems for our pair of dual models. Nanda and Das' results (Optimization 28 (1994) 267; Eur. J. Oper. Res. 88 (1996) 572) are obtained as special cases.