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1.
《Optimization》2012,61(4):401-412
The aim of this article is to study the relationship between generalized Minty vector variational inequalities and non-smooth vector optimization problems. Under pseudoconvexity or pseudomonotonicity, we establish the relationship between an efficient solution of a non-smooth vector optimization problem and a generalized Minty vector variational inequality. This offers a non-smooth version of existing Minty variational principle. 相似文献
2.
The vector optimization problem may have a nonsmooth objective function. Therefore, we introduce the Minty vector variational
inequality (Minty VVI) and the Stampacchia vector variational inequality (Stampacchia VVI) defined by means of upper Dini
derivative. By using the Minty VVI, we provide a necessary and sufficient condition for a vector minimal point (v.m.p.) of
a vector optimization problem for pseudoconvex functions involving Dini derivatives. We establish the relationship between
the Minty VVI and the Stampacchia VVI under upper sign continuity. Some relationships among v.m.p., weak v.m.p., solutions
of the Stampacchia VVI and solutions of the Minty VVI are discussed. We present also an existence result for the solutions
of the weak Minty VVI and the weak Stampacchia VVI. 相似文献
3.
Some Remarks on the Minty Vector Variational Inequality 总被引:4,自引:0,他引:4
Yang X. M. Yang X. Q. Teo K. L. 《Journal of Optimization Theory and Applications》2004,121(1):193-201
In this paper, we establish some relations between a Minty vector variational inequality and a vector optimization problem under pseudoconvexity or pseudomonotonicity, respectively. Our results generalize those of Ref. 1. 相似文献
4.
The aim of this paper is to study the relationship among Minty vector variational-like inequality problem, Stampacchia vector variational-like inequality problem and vector optimization problem involving (G, α)-invex functions. Furthermore, we establish equivalence among the solutions of weak formulations of Minty vector variational-like inequality problem, Stampacchia vector variational-like inequality problem and weak efficient solution of vector optimization problem under the assumption of (G, α)-invex functions. Examples are provided to elucidate our results. 相似文献
5.
G. P. Crespi A. Guerraggio M. Rocca 《Journal of Optimization Theory and Applications》2007,132(1):213-226
In this paper, we give notions of well posedness for a vector optimization problem and a vector variational inequality of
the differential type. First, the basic properties of well-posed vector optimization problems are studied and the case of
C-quasiconvex problems is explored. Further, we investigate the links between the well posedness of a vector optimization problem
and of a vector variational inequality. We show that, under the convexity of the objective function f, the two notions coincide. These results extend properties which are well known in scalar optimization.
Communicated by F. Giannessi 相似文献
6.
In this paper, we study the relationship among the generalized Minty vector variational-like inequality problem, generalized
Stampacchia vector variational-like inequality problem and vector optimization problem for nondifferentiable and nonconvex
functions. We also consider the weak formulations of the generalized Minty vector variational-like inequality problem and
generalized Stampacchia vector variational-like inequality problem and give some relationships between the solutions of these
problems and a weak efficient solution of the vector optimization problem. 相似文献
7.
Vivek Laha 《Optimization》2017,66(11):1837-1850
In this paper, we establish some results which exhibit an application of convexificators in vector optimization problems (VOPs) and vector variational inequaities involving locally Lipschitz functions. We formulate vector variational inequalities of Stampacchia and Minty type in terms of convexificators and use these vector variational inequalities as a tool to find out necessary and sufficient conditions for a point to be a vector minimal point of the VOP. We also consider the corresponding weak versions of the vector variational inequalities and establish several results to find out weak vector minimal points. 相似文献
8.
9.
In this paper, we introduce a Minty type vector variational inequality, a Stampacchia type vector variational inequality, and the weak forms of them, which are all defined by means of subdifferentials on Hadamard manifolds. We also study the equivalent relations between the vector variational inequalities and nonsmooth convex vector optimization problems. By using the equivalent relations and an analogous to KKM lemma, we give some existence theorems for weakly efficient solutions of convex vector optimization problems under relaxed compact assumptions. 相似文献
10.
X. Q. Yang 《Journal of Optimization Theory and Applications》1997,95(3):729-734
The study of a vector variational inequality has been advanced because it has many applications in vector optimization problems and vector equilibrium flows. In this paper, we discuss relations between a solution of a vector variational inequality and a Pareto solution or a properly efficient solution of a vector optimization problem. We show that a vector variational inequality is a necessary and sufficient optimality condition for an efficient solution of the vector pseudolinear optimization problem. 相似文献
11.
This paper is intended to study the vector variational inequalities on Hadamard manifolds. Generalized Minty and Stampacchia vector variational inequalities are introduced involving generalized subdifferential. Under strongly geodesic convexity, relations between solutions of these inequalities and a nonsmooth vector optimization problem are established. To illustrate the relationship between a solution of generalized weak Stampacchia vector variational inequality and weak efficiency of a nonsmooth vector optimization problem, a non-trivial example is presented. 相似文献
12.
《Optimization》2012,61(3):327-338
In this article, two types of cone-pseudomonotone bifunctions have been introduced and the weaker form of pseudomonotonicity is used to establish an existence theorem for a Stampacchia-kind vector variational inequality problem given in terms of bifunctions. For a vector optimization problem, the necessary and sufficient optimality conditions in terms of an associated vector variational inequality problem have been established using a generalized form of cone pseudoconvexity of objective function. 相似文献
13.
In optimization, objective functions which are both pseudoconvex and pseudoconcave have been studied extensively. Generalizing these results, we characterize pseudomonotone maps F where -F is also pseudomonotone and explore their properties in variational inequality problems. In particular, we extend recent results by Jeyakumar and Yang which were derived for optimization problems. 相似文献
14.
On Nondifferentiable and Nonconvex Vector Optimization Problems 总被引:5,自引:0,他引:5
In this paper, we prove the equivalence among the Minty vector variational-like inequality, Stampacchia vector variational-like inequality, and a nondifferentiable and nonconvex vector optimization problem. By using a fixed-point theorem, we establish also an existence theorem for generalized weakly efficient solutions to the vector optimization problem for nondifferentiable and nonconvex functions. 相似文献
15.
Giovanni P. Crespi Andreas H. Hamel Carola Schrage 《Journal of Mathematical Analysis and Applications》2015
Extremal problems are studied involving an objective function with values in (order) complete lattices of sets generated by so-called set relations. Contrary to the popular paradigm in vector optimization, the solution concept for such problems, introduced by F. Heyde and A. Löhne, comprises the attainment of the infimum as well as a minimality property. The main result is a Minty type variational inequality for set optimization problems which provides a sufficient optimality condition under lower semicontinuity assumptions and a necessary condition under appropriate generalized convexity assumptions. The variational inequality is based on a new Dini directional derivative for set-valued functions which is defined in terms of a “lattice difference quotient.” A residual operation in a lattice of sets replaces the inverse addition in linear spaces. Relationships to families of scalar problems are pointed out and used for proofs. The appearance of improper scalarizations poses a major difficulty which is dealt with by extending known scalar results such as Diewert's theorem to improper functions. 相似文献
16.
In this paper, we study a generalized weak vector variational inequality, which is a generalization of a weak vector variational inequality and a Minty weak vector variational inequality. By virtue of a contingent derivative and a Φ-contingent cone, we investigate differential properties of a class of set-valued maps and obtain an explicit expression of its contingent derivative. We also establish some necessary optimality conditions for solutions of the generalized weak vector variational inequality, which generalize the corresponding results in the literature. Furthermore, we establish some unified necessary and sufficient optimality conditions for local optimal solutions of the generalized weak vector variational inequality. Simultaneously, we also show that there is no gap between the necessary and sufficient conditions under an appropriate condition. 相似文献
17.
在拓扑向量空间中讨论下Dini方向导数形式的广义Minty向量似变分不等式问题. 可微形式的Minty变分不等式、Minty似变分不等式和Minty向量变分不等式是其特殊形式. 该文分别讨论了Minty向量似变分不等式的解与径向递减函数, 与向量优化问题的最优解或有效解之间的关系问题, 以及Minty向量似变分不等式的解集的仿射性质. 这些定理推广了文献中Minty变分不等式的一些重要的已知结果. 相似文献
18.
On minty variational principle for nonsmooth vector optimization problems with approximate convexity
In this paper, we consider a vector optimization problem involving locally Lipschitz approximately convex functions and give several concepts of approximate efficient solutions. We formulate approximate vector variational inequalities of Stampacchia and Minty type and use these inequalities as a tool to characterize an approximate efficient solution of the vector optimization problem. 相似文献
19.
Qamrul Hasan Ansari Mahboubeh Rezaie Jafar Zafarani 《Journal of Global Optimization》2012,53(2):271-284
In this paper, we consider different kinds of generalized vector variational-like inequality problems and a vector optimization problem. We establish some relationships between the solutions of generalized Minty vector variational-like inequality problem and an efficient solution of a vector optimization problem. We define a perturbed generalized Stampacchia vector variational-like inequality problem and discuss its relation with generalized weak Minty vector variational-like inequality problem. We establish some existence results for solutions of our generalized vector variational-like inequality problems. 相似文献