A new Fenchel dual problem in vector optimization

We introduce a new Fenchel dual for vector optimization problems inspired by the form of the Fenchel dual attached to the scalarized primal multiobjective problem. For the vector primal-dual pair we prove weak and strong duality. Furthermore, we recall two other Fenchel-type dual problems introduced in the past in the literature, in the vector case, and make a comparison among all three duals. Moreover, we show that their sets of maximal elements are equal.     

A projection iterative algorithm for strong vector equilibrium problem

In this paper, iterative algorithm for strong vector equilibrium problem (SVEP) is studied. Firstly, an auxiliary problem for SVEP is introduced and the relationships between these two problems are discussed. Then, based on the auxiliary problem, a projection iterative algorithm for SVEP is proposed. Moreover, analysis of convergence of this iterative algorithm is investigated under suitable conditions of continuity and convexity. The main result obtained in this paper generalizes and improves the corresponding ones of Iusem and Sosa [Iusem AN, Sosa W. Iterative algorithms for equilibrium problems. Optimization. 2003;52(3):301–316.] and Cheng and Liu [Cheng B, Liu SY. An iterative algorithm for vector equilibrium problems. J. Lanzhou Univ. (Nat. Sci.). 2009;45(5):105–109.].     

A remark on a standard and linear vector network equilibrium problem with capacity constraints

In this paper, (weak) vector equilibrium principle with capacity constraints is introduced. A necessary condition that a vector minimum cost flow is a vector equilibrium flow with capacity constraints is obtained. When the number of paths connecting with each pair of source and sink is less than or equal to 2, a sufficient condition for a vector minimum cost flow to be a vector equilibrium flow is also obtained. A generalized (weak) vector equilibrium principle is also introduced. Without any additional assumption, a necessary and sufficient condition for a (weak) vector minimum cost flow to be a generalized (weak) vector equilibrium flow is obtained.     

Vector equilibrium problems with elastic demands and capacity constraints

In this paper, a (weak) vector equilibrium principle for vector network problems with capacity constraints and elastic demands is introduced. A sufficient condition for a (weak) vector equilibrium flow to be a solution for a system of (weak) vector quasi-variational inequalities is obtained. By virtue of Gerstewitz’s nonconvex separation functional ξ, a (weak) ξ-equilibrium flow is introduced. Relations between a weak vector equilibrium flow and a (weak) ξ-equilibrium flow is investigated. Relations between weak vector equilibrium flows and two classes of variational inequalities are also studied.     

Continuity of the Solution Set to Parametric Weak Vector Equilibrium Problems

In this paper, we obtain some stability results for parametric weak vector equilibrium problems in topological vector spaces. We provide sufficient conditions for the continuity of the solution set mapping in parametric weak monotone vector equilibrium problems. This research was partially supported by the National Natural Science Foundation of China (Grant 10561007) and the Natural Science Foundation of Jiangxi Province, China.     

关于Aluthge变换的数值域

设A是作用在希耳伯特空间H上的有界线性算子,如果A=V A是算子A的极分解,则定义A~=A 12V A 21和A~(*)=A*21V A*21分别为算子A的Aluthge变换A~和*-Aluthge变换A~(*).记A~和A~(*)的数值域分别为W(A~)和W(A~(*)).证明了W(A~)=W(A~(*)),即肯定了吴提出的一个猜想.     

The multiparameter eigenvalue problem: Jordan vector semilattices

In this paper, multiparameter eigenvalue (MPE) problems for matrices are considered. The notion of Jordan vector semilattices as a generalization of the notion of Jordan vector chains is introduced for a multiple spectrum point of disconnected MPE problems. The notion of generating vector is introduced. For the linear case, a special form of equations determining Jordan vector semilattices is presented. The above notions are extended to the case of connected MPE problems, including linear ones. The relationship between the Jordan vector semilattices of a connected linear MPE problem and the Jordan vector chains of the corresponding simultaneous spectral problems for matrices is established. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 219, 1994, pp. 213–220. This paper was supported by the Russian Foundation of Fundamental Research (grant 94-01-00919). Translated by V. B. Khazanov.     

Strong vector F-complementary problem and least element problem of feasible set

In this paper we introduce several classes of strong vector F-complementary problems and give some existence results for these problems in Banach spaces. We also discuss the least element problems of feasible sets and present their relations with the strong vector F-complementary problems.     

Weak Subdifferential of Set-Valued Mappings

In this note we introduce a notion of the weak contingent generalized gradient for set-valued mappings associated with the contingent epiderivative of set-valued mappings introduced in "E. Bednarczuk and W. Song (1998). Contingent epiderivative and its applications to set-valued optimization. Control and Cybernetics, 27, 376-386; G.Y. Chen and J. Jahn (1998). Optimally conditions for set-valued optimization problems. Mathematical Methods of Operations Research, 48, 187-200." and prove that, under some additional condition, it coincides with the weak subdifferential introduced in "T. Tanino (1992). Conjugate duality in vector optimization. Journal of Mathematical Analysis and Applications, 167, 84-97." when the set-valued map is cone-convex. We also study the weak contingent generalized gradient of a sum of two set-valued mappings and optimality conditions for a set-valued vector optimization problem.     

Some relations between vector variational inequality problems and nonsmooth vector optimization problems using quasi efficiency

This paper deals with the relations between vector variational inequality problems and nonsmooth vector optimization problems using the concept of quasi efficiency. We identify the vector critical points, the weak quasi efficient points and the solutions of the weak vector variational inequality problems under generalized approximate convexity assumptions. To the best of our knowledge such results have not been established till now.     

On vector optimization problems and vector variational inequalities using convexificators

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.     

Gap functions and error bounds for nonsmooth convex vector optimization problem

Abstract

In this article, our main aim is to develop gap functions and error bounds for a (non-smooth) convex vector optimization problem. We show that by focusing on convexity we are able to quite efficiently compute the gap functions and try to gain insight about the structure of set of weak Pareto minimizers by viewing its graph. We will discuss several properties of gap functions and develop error bounds when the data are strongly convex. We also compare our results with some recent results on weak vector variational inequalities with set-valued maps, and also argue as to why we focus on the convex case.     

Stability results for properly quasi convex vector optimization problems

In this paper, we discuss the stability of the sets of (weak) minimal points and (weak) efficient points of vector optimization problems. Assuming that the objective functions are (strictly) properly quasi convex, and the data ofthe approximate problems converges to the data of the original problems in the sense of Painlevé–Kuratowski, we establish the Painlevé–Kuratowski set convergence of the sets of (weak) minimal points and (weak) efficient points of the approximate problems to the corresponding ones of original problem. Our main results improve and extend the results of the recent papers.     

On generalized vector equilibrium problems

In this paper, we study the generalized vector equilibrium problems in real Hausdorff topological vector space settings. The concepts of weak solutions and strong solutions are introduced. Several new results of existence for the weak solutions and strong solutions of generalized vector equilibrium problems are derived. The new results extend and modify various existence theorems for similar problems.     

向量拟平衡问题的本质解及解集的本质连通区

本文研究向量拟平衡问题,得到了向量拟平衡问题解的一个存在性结果,证明了在满足一定的连续性和凸性条件的问题构成的空间Y中,大多数(在Baire分类意义下)问题的解集是稳定的,并证明Y的某子集中,每个向量拟平衡问题的解集中至少存在一个本质连通区。作为应用,我们导出了多目标广义对策弱Pareto-Nash平衡点的存在性,证明了在满足一定的连续性和凸性条件的多目标广义对策构成的空间P中,大多数对策的弱Pareto-Nash平衡点是稳定的,并证明了P中的每个对策的弱Pareto-Nash平衡点集中至少有一个本质连通区。     

Inner and outer estimates for the solution sets and their asymptotic cones in vector optimization

We use asymptotic analysis to develop finer estimates for the efficient, weak efficient and proper efficient solution sets (and for their asymptotic cones) to convex/quasiconvex vector optimization problems. We also provide a new representation for the efficient solution set without any convexity assumption, and the estimates involve the minima of the linear scalarization of the original vector problem. Some new necessary conditions for a point to be efficient or weak efficient solution for general convex vector optimization problems, as well as for the nonconvex quadratic multiobjective optimization problem, are established.     

The Stability of Set of Solutions for Symmetric Vector Quasi-Equilibrium Problems

In this paper, the stability of the set of solutions for symmetric vector quasi-equilibrium problems is discussed. Then, we prove a generic stability theorem and give an existence theorem for essentially connected components of the set of solutions for symmetric vector quasi-equilibrium problems. Finally, we apply these results to vector weak saddle point problems with constraints. This research was partially supported by the Natural Science Foundation of China (Grant number: 10561007) and Natural Science Foundation of Jiangxi Province, China.     

Vector duality for convex vector optimization problems by means of the quasi-interior of the ordering cone

We define weakly minimal elements of a set with respect to a convex cone by means of the quasi-interior of the cone and characterize them via linear scalarization, generalizing the classical weakly minimal elements from the literature. Then we attach to a general vector optimization problem, a dual vector optimization problem with respect to (generalized) weakly efficient solutions and establish new duality results. By considering particular cases of the primal vector optimization problem, we derive vector dual problems with respect to weakly efficient solutions for both constrained and unconstrained vector optimization problems and the corresponding weak, strong and converse duality statements.     

A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization

In this paper, we introduce a new iterative scheme for finding a common element of the set of solutions of an equilibrium problem, the set of common fixed point for a family of infinitely nonexpansive mappings and the set of solutions of the variational inequality for α$\alpha$-inverse-strongly monotone mappings in a Hilbert space. Under suitable conditions, some strong convergence theorems for approximating a common element of the above three sets are obtained. As applications, at the end of the paper we utilize our results to study the optimization problem and some convergence problem for strictly pseudocontractive mappings. The results presented in the paper extend and improve some recent results of Yao and Yao [Y.Y. Yao, J.C. Yao, On modified iterative method for nonexpansive mappings and monotone mappings, Appl. Math. Comput. 186 (2) (2007) 1551–1558], Plubtieng and Punpaeng [S. Plubtieng, R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonlinear mappings and monotone mappings, Appl. Math. Comput. (2007) doi:10.1016/j.amc.2007.07.075], S. Takahashi and W. Takahashi [S. Takahashi, W. Takahashi, Viscosity approximation methods for Equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2006) 506–515], Su, Shang and Qin [Y.F. Su, M.J. Shang, X.L. Qin, An iterative method of solution for equilibrium and optimization problems, Nonlinear Anal. (2007) doi:10.1016/j.na.2007.08.045] and Chang, Cho and Kim [S.S. Chang, Y.J. Cho, J.K. Kim, Approximation methods of solutions for equilibrium problem in Hilbert spaces, Dynam. Systems Appl. (in print)].     

Vector variational-like inequalities and non-smooth vector optimization problems

In this paper, we establish relationships between vector variational-like inequality problems and non-smooth vector optimization problems under non-smooth invexity. We identify the vector critical points, the weakly efficient points and the solutions of the non-smooth weak vector variational-like inequality problems, under non-smooth pseudo-invexity assumptions. These conditions are more general than those existing in the literature.