Optimality conditions and duality in subdifferentiable multiobjective fractional programming

Fritz John and Kuhn-Tucker necessary and sufficient conditions for a Pareto optimum of a subdifferentiable multiobjective fractional programming problem are derived without recourse to an equivalent convex program or parametric transformation. A dual problem is introduced and, under convexity assumptions, duality theorems are proved. Furthermore, a Lagrange multiplier theorem is established, a vector-valued ratio-type Lagrangian is introduced, and vector-valued saddle-point results are presented.The authors are thankful to the referees and Professor P. L. Yu for their many useful comments and suggestions which have improved the presentation of the paper.The first author is thankful to the Natural Science and Engineering Research Council of Canada for financial support through Grant No. A-5319. The authors are also thankful to the Dean's Office, Faculty of Management, University of Manitoba, for the financial support provided for the third author's visit to the Faculty.     

带消失约束的区间值优化问题的最优性条件与对偶定理

本文考虑一类带消失约束的非光滑区间值优化问题(IOPVC)。在一定的约束条件下得到了问题(IOPVC)的LU最优解的必要和充分性最优性条件,研究了其与Mond-Weir型对偶模型和Wolfe型对偶模型之间的弱对偶,强对偶和严格逆对偶定理,并给出了一些例子来阐述我们的结果。     

Second-order optimality conditions using approximations for nonsmooth vector optimization problems under inclusion constraints

Second-order necessary conditions and sufficient conditions for optimality in nonsmooth vector optimization problems with inclusion constraints are established. We use approximations as generalized derivatives and avoid even continuity assumptions. Convexity conditions are not imposed explicitly. Not all approximations in use are required to be bounded. The results improve or include several recent existing ones. Examples are provided to show that our theorems are easily applied in situations where several known results do not work.     

Optimality conditions and duality for nondifferentiable multiobjective programming problems involving d-r-type I functions

In this paper, new classes of nondifferentiable functions constituting multiobjective programming problems are introduced. Namely, the classes of d$d$-r$r$-type I objective and constraint functions and, moreover, the various classes of generalized d$d$-r$r$-type I objective and constraint functions are defined for directionally differentiable multiobjective programming problems. Sufficient optimality conditions and various Mond–Weir duality results are proved for nondifferentiable multiobjective programming problems involving functions of such type. Finally, it is showed that the introduced d$d$-r$r$-type I notion with r≠0$r\ne 0$ is not a sufficient condition for Wolfe weak duality to hold. These results are illustrated in the paper by suitable examples.     

Optimality conditions and duality for a class of nondifferentiable multiobjective generalized fractional programming problems

This paper studies a class of multiobjective generalized fractional programming problems, where the numerators of objective functions are the sum of differentiable function and convex function, while the denominators are the difference of differentiable function and convex function. Under the assumption of Calmness Constraint Qualification the Kuhn-Tucker type necessary conditions for efficient solution are given, and the Kuhn-Tucker type sufficient conditions for efficient solution are presented under the assumptions of （F, α, ρ, d）-V-convexity. Subsequently, the optimality conditions for two kinds of duality models are formulated and duality theorems are proved.     

The Karush-Kuhn-Tucker optimality conditions for the optimization problem with fuzzy-valued objective function

The Karush-Kuhn-Tucker (KKT) conditions for an optimization problem with fuzzy-valued objective function are derived in this paper. A solution concept of this optimization problem is proposed by considering an ordering relation on the class of all fuzzy numbers. The solution concept proposed in this paper will follow from the similar solution concept, called non-dominated solution, in the multiobjective programming problem. In order to consider the differentiation of a fuzzy-valued function, we use the Hausdorff metric to define the distance between two fuzzy numbers and the Hukuhara difference to define the difference of two fuzzy numbers. Under these settings, the KKT optimality conditions are elicited naturally by introducing the Lagrange function multipliers.     

Optimality and Duality for Invex Nonsmooth Multiobjective programming problems

We consider nonsmooth multiobjective programming problems with inequality and equality constraints involving locally Lipschitz functions. Several sufficient optimality conditions under various (generalized) invexity assumptions and certain regularity conditions are presented. In addition, we introduce a Wolfe-type dual and Mond–Weir-type dual and establish duality relations under various (generalized) invexity and regularity conditions.     

具有(F α,ρ,d)-V-凸的非光滑多目标分式规划的最优性条件和对偶性

本文考虑了一类非光滑多目标分式规划问题,该多目标分式规划问题中所出现的函数是局部Lipschitz的．对该类多目标分式规划问题,引入了(F,α,ρ,d)-V-凸函数的概念,证明了有效解的充分条件和必要条件,构造出了一种参数对偶模型和一种半参数对偶模型,并证明了相应的对偶定理．     

Extended Farkas lemma and strong duality for composite optimization problems with DC functions

In this paper, we consider the optimization problem in locally convex Hausdorff topological vector spaces with objectives given as the difference of two composite functions and constraints described by an arbitrary (possibly infinite) number of convex inequalities. Using the epigraph technique, we introduce some new constraint qualifications, which completely characterize the Farkas lemma, the dualities between the primal problem and its dual problem. Applications to the conical programming with DC composite function are also given.