广义凸多目标规划的Mond─Weir型对偶性

本文研究带不等式和等式约束的多目标规划的Ｍｏｎｄ－Ｗｅｉｒ型对偶性理论。在目标和约束是广义凸的假设下，证明了弱对偶定理、直接对偶定理以及逆对偶定理     

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.     

一类多目标分式规划的二阶对称对偶问题

研究一类多目标分式规划的二阶对称对偶问题.在二阶F-凸性假设下给出了对偶问题的弱对偶、强对偶和逆对偶定理.并在对称和反对称假设下研究了该问题的自身对偶性.     

Duality in nonlinear fractional programming

Summary The purpose of the present paper is to introduce, on the lines similar to that ofWolfe [1961], a dual program to a nonlinear fractional program in which the objective function, being the ratio of a convex function to a strictly positive linear function, is a special type of pseudo-convex function and the constraint set is a convex set constrained by convex functions in the form of inequalities. The main results proved are, (i) Weak duality theorem, (ii)Wolfe's (Direct) duality theorem and (iii)Mangasarian's Strict Converse duality theorem.Huard's [1963] andHanson's [1961] converse duality theorems for the present problem have just been stated because they can be obtained as a special case ofMangasarian's theorem [1969, p. 157]. The other important discussion included is to show that the dual program introduced in the present paper can also be obtained throughDinkelbach's Parametric Replacement [1967] of a nonlinear fractional program. Lastly, duality in convex programming is shown to be a special case of the present problem.The present research is partially supported by National Research Council of Canada.     

Nondifferentiable multiobjective Mond–Weir type second-order symmetric duality over cones

In this paper, a pair of Mond–Weir type nondifferentiable multiobjective second-order symmetric dual programs over arbitrary cones is formulated. Weak, strong and converse duality theorems are established under second-order K–F-convexity/K–η-bonvexity assumptions. A self duality theorem is also obtained by assuming the functions involved to be skew-symmetric.     

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.     

超有效意义下向量集值优化修整的Lagrange乘子型对偶

给出了一类加细的向量集值优化超有效解的最优性条件,由此给出了一种改进的Lagrange乘子型对偶,并建立了对偶的弱定理,正定理及逆定理。     

A generalization of the Gronwall-Bellman lemma and its applications

In this study we present an important theorem of the alternative involving convex functions and convex cones. From this theorem we develop saddle value optimality criteria and stationary optimality criteria for convex programs. Under suitable constraint qualification we obtain a generalized form of the Kuhn-Tucker conditions. We also use the theorem of the alternative in developing an important duality theorem. No duality gaps are encountered under the constraint qualification imposed earlier and the dual problem always possesses a solution. Moreover, it is shown that all constraint qualifications assure that the primal problem is stable in the sense used by Gale and others. The notion of stability is closely tied up with the positivity of the lagrangian multiplier of the objective function.     

A class of nondifferentiable continuous programming problems

Optimality conditions, duality and converse duality results are obtained for a class of continuous programming problems with a nondifferentiable term in the integrand of the objective function. The proofs are based on a Fritz John theorem for constrained optimization in abstract spaces. The results generalize various well-known results in variational problems with differentiable functions, and also give a dynamic analogue of certain nondifferentiable programming problems.     

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。     

广义I型连通级小极大分式问题的对偶

在I型弧连通和广义I型弧连通假设下,建立了极大极小分式优化问题的对偶模型,并提出了弱对偶定理、强对偶定理和严格逆对偶定理.     

Duality on a nondifferentiable minimax fractional programming

We establish the necessary and sufficient optimality conditions on a nondifferentiable minimax fractional programming problem. Subsequently, applying the optimality conditions, we constitute two dual models: Mond-Weir type and Wolfe type. On these duality types, we prove three duality theorems??weak duality theorem, strong duality theorem, and strict converse duality theorem.     

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.     

An extension lemma and homogeneous programming

An extension lemma, which is equivalent to the generalized Gordan's theorem of the alternative, due to Fan, Glicksberg, and Hoffman, is applied to present a duality theory for a general class of homogeneous programs, with and without a constraint qualification of Slater type. In addition, an existence theorem for optimal solutions of homogeneous programs is given.The author thanks an anonymous referee for valuable suggestions about an earlier draft of this paper.     

Nondifferentiable mathematical programming and convex-concave functions

For a convex-concave functionL(x, y), we define the functionf(x) which is obtained by maximizingL with respect toy over a specified set. The minimization problem with objective functionf is considered. We derive necessary conditions of optimality for this problem. Based upon these necessary conditions, we define its dual problem. Furthermore, a duality theorem and a converse duality theorem are obtained. It is made clear that these results are extensions of those derived in studies on a class of nondifferentiable mathematical programming problems.This work was supported by the Japan Society for the Promotion of Sciences.     

Regularized Lagrangian duality for linearly constrained quadratic optimization and trust-region problems

In this paper we first establish a Lagrange multiplier condition characterizing a regularized Lagrangian duality for quadratic minimization problems with finitely many linear equality and quadratic inequality constraints, where the linear constraints are not relaxed in the regularized Lagrangian dual. In particular, in the case of a quadratic optimization problem with a single quadratic inequality constraint such as the linearly constrained trust-region problems, we show that the Slater constraint qualification (SCQ) is necessary and sufficient for the regularized Lagrangian duality in the sense that the regularized duality holds for each quadratic objective function over the constraints if and only if (SCQ) holds. A new theorem of the alternative for systems involving both equality constraints and two quadratic inequality constraints plays a key role. We also provide classes of quadratic programs, including a class of CDT-subproblems with linear equality constraints, where (SCQ) ensures regularized Lagrangian duality.     

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,…     

More on subgradient duality

A duality theorem of P. Wolfe for nonlinear differential programming has been extended by the author to the non-differentiable case by replacing gradients by subgradients. In this paper this extended result is improved by allowing additional types of constraints. Also a converse duality theorem is proved.     

Duality Theorems for Separable Convex Programming Without Qualifications

In the research of mathematical programming, duality theorems are essential and important elements. Recently, Lagrange duality theorems for separable convex programming have been studied. Tseng proves that there is no duality gap in Lagrange duality for separable convex programming without any qualifications. In other words, although the infimum value of the primal problem equals to the supremum value of the Lagrange dual problem, Lagrange multiplier does not always exist. Jeyakumar and Li prove that Lagrange multiplier always exists without any qualifications for separable sublinear programming. Furthermore, Jeyakumar and Li introduce a necessary and sufficient constraint qualification for Lagrange duality theorem for separable convex programming. However, separable convex constraints do not always satisfy the constraint qualification, that is, Lagrange duality does not always hold for separable convex programming. In this paper, we study duality theorems for separable convex programming without any qualifications. We show that a separable convex inequality system always satisfies the closed cone constraint qualification for quasiconvex programming and investigate a Lagrange-type duality theorem for separable convex programming. In addition, we introduce a duality theorem and a necessary and sufficient optimality condition for a separable convex programming problem, whose constraints do not satisfy the Slater condition.     

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.