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.     

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.     

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.     

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.     

Multiobjective Variational Programming under Generalized Type I Univexity

In this paper, we extend the Mishra, Rueda and Giorgi generalized V-univexity type I, defined for multiobjective programming, to multiobjective variational programming problems and we derive various sufficient optimality conditions and mixed type duality results under generalized V-univexity type I conditions. The authors thank the referee for many valuable comments and helpful suggestions.     

Generalized vector variational inequality and fuzzy extension

A generalized vector variational inequality (GVVI) is considered. We establish the existence theorem for (GVVI) under assumptions of C-pseudomonotonicity and V-hemicontinuity. From our existence theorem, we obtain the fuzzy extension of a result of Chen and Yang.     

Minimax and symmetric duality for nonlinear multiobjective mixed integer programming

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

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.     

含有BF—I函数的多目标变分问题的混合对偶性

在有效解的意义下,对一类含有BF—I函数的多目标变分问题给出了混合型对偶的强对偶定理、弱对偶定理和严格逆对偶定理。     

η-Approximation Method for Non-convex Multiobjective Variational Problems

In this paper, we use the η-approximation method for a class of non-convex multiobjective variational problems with invex functionals. In this approach, for the considered multiobjective variational problem, the associated η-approximated multiobjective variational problem is constructed at the given feasible solution. The equivalence between (weakly) e?cient solutions in the original multiobjective variational problem and its associated η-approximated multiobjective variational problem is established under invexity hypotheses.     

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.     

Image Space Analysis for Constrained Inverse Vector Variational Inequalities via Multiobjective Optimization

In this paper, we employ the image space analysis to study constrained inverse vector variational inequalities. First, sufficient and necessary optimality conditions for constrained inverse vector variational inequalities are established by using multiobjective optimization. A continuous nonlinear function is also introduced based on the oriented distance function and projection operator. This function is proven to be a weak separation function and a regular weak separation function under different parameter sets. Then, two alternative theorems are established, which lead directly to sufficient and necessary optimality conditions of the inverse vector variational inequalities. This provides a partial answer to an open question posed in Chen et al. (J Optim Theory Appl 166:460–479, 2015).     

Methods of variational analysis in multiobjective optimization

This article studies new applications of advanced methods of variational analysis and generalized differentiation to constrained problems of multiobjective/vector optimization. We pay most attention to general notions of optimal solutions for multiobjective problems that are induced by geometric concepts of extremality in variational analysis, while covering various notions of Pareto and other types of optimality/efficiency conventional in multiobjective optimization. Based on the extremal principles in variational analysis and on appropriate tools of generalized differentiation with well-developed calculus rules, we derive necessary optimality conditions for broad classes of constrained multiobjective problems in the framework of infinite-dimensional spaces. Applications of variational techniques in infinite dimensions require certain ‘normal compactness’ properties of sets and set-valued mappings, which play a crucial role in deriving the main results of this article.     

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.     

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.     

Robust optimization for interactive multiobjective programming with imprecise information applied to R&D project portfolio selection

A multiobjective binary integer programming model for R&D project portfolio selection with competing objectives is developed when problem coefficients in both objective functions and constraints are uncertain. Robust optimization is used in dealing with uncertainty while an interactive procedure is used in making tradeoffs among the multiple objectives. Robust nondominated solutions are generated by solving the linearized counterpart of the robust augmented weighted Tchebycheff programs. A decision maker’s most preferred solution is identified in the interactive robust weighted Tchebycheff procedure by progressively eliciting and incorporating the decision maker’s preference information into the solution process. An example is presented to illustrate the solution approach and performance. The developed approach can also be applied to general multiobjective mixed integer programming problems.     

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 .     

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].     

Sufficiency and Duality in Multiobjective Variational Problems with Generalized Type I Functions

Recently Hachimi and Aghezzaf introduced the notion of (F,α,ρ,d)-type I functions, a new class of functions that unifies several concepts of generalized type I functions. Here, we extend the concepts of (F,α,ρ,d)-type I and generalized (F,α,ρ, d)-type I functions to the continuous case and we use these concepts to establish various sufficient optimality conditions and mixed duality results for multiobjective variational problems. Our results apparently generalize a fairly large number of sufficient optimality conditions and duality results previously obtained for multiobjective variational problems.     

一种新的向量互补问题

本文在实局部凸空间中引入了一种新的向量互补问题,这一向量互补问题不仅包含了由Ｙｕ和Ｙａｏ提出的广义向量互补问题由Ｃｈｅｎ和Ｙａｎｇ定义的弱向量互补问题,而且还包含了Ｉｓａｃ意义下的隐互补问题。本文还讨论了新的向量互补问题,向量变分不等式,向量单向极小化问题和最小元问题之间的关系,给出了这一向量互补问题解的存在定理。