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1.
A Nonlinear Scalarization Function and Generalized Quasi-vector Equilibrium Problems 总被引:1,自引:0,他引:1
Scalarization method is an important tool in the study of vector optimization as corresponding solutions of vector optimization
problems can be found by solving scalar optimization problems. In this paper we introduce a nonlinear scalarization function
for a variable domination structure. Several important properties, such as subadditiveness and continuity, of this nonlinear
scalarization function are established. This nonlinear scalarization function is applied to study the existence of solutions
for generalized quasi-vector equilibrium problems.
This paper is dedicated to Professor Franco Giannessi for his 68th birthday 相似文献
2.
In this paper, a nonlinear scalarization function is introduced for a variable domination structure. It is shown that this function is positively homogeneous, subadditive, and strictly monotone. This nonlinear function is then applied to characterize the weakly nondominated solution of multicriteria decision making problems and the solution of vector variational inequalities. 相似文献
3.
4.
We study the weak domination property and weakly efficient solutions in vector optimization problems. In particular scalarization
of these problems is obtained by virtue of some suitable merit functions. Some natural conditions to ensure the existence
of error bounds for merit functions are also given.
This research was supported by a direct grant (CUHK) and an Earmarked Grant from the Research Grant Council of Hong Kong. 相似文献
5.
《Optimization》2012,61(2):305-319
The scalarization functions were used in vector optimization for a long period. Similar functions were introduced and used in economics under the name of shortage function or in mathematical finance under the name of (convex or coherent) measures of risk. The main aim of this article is to study Lipschitz continuity properties of such functions and to give some applications for deriving necessary optimality conditions for vector optimization problems using the Mordukhovich subdifferential. 相似文献
6.
In vector optimization with a variable ordering structure, the partial ordering defined by a convex cone is replaced by a whole family of convex cones, one associated with each element of the space. In recent publications, it was started to develop a comprehensive theory for these vector optimization problems. Thereby, also notions of proper efficiency were generalized to variable ordering structures. In this paper, we study the relation between several types of proper optimality. We give scalarization results based on new functionals defined by elements from the dual cones which allow complete characterizations also in the nonconvex case. 相似文献
7.
Behnam Soleimani 《Journal of Optimization Theory and Applications》2014,162(2):605-632
In this paper, we deal with approximate solutions in vector-optimization problems with respect to a variable order structure. In the case of exact solutions of a vector optimization problem, especially in the variable order case, authors use a cone or a pointed convex cone-valued map in order to describe the solution concepts but in this paper, we use a set-valued map and this map is not a (pointed convex) cone-valued map necessarily. We characterize these solution concepts by a general scalarization method by means of nonlinear functionals. In the last section, an extension of Ekeland’s variational principle for a vector optimization problem with a variable order structure is given. 相似文献
8.
推广固定锥序下的改进集概念到变动序关系.引入了带变动序结构的向量优化问题的E-最优元.应用Tammer-Weidner意义下的非线性标量化函数,给出了向量优化问题E-最优元的标量化刻画,建立了带变动序结构的向量优化问题的E-最优元的必要和充分最优性条件. 相似文献
9.
In this paper, we develop a method of study of Levitin?CPolyak well-posedness notions for vector valued optimization problems using a class of scalar optimization problems. We first introduce a non-linear scalarization function and consider its corresponding properties. We also introduce the Furi?CVignoli type measure and Dontchev?CZolezzi type measure to scalar optimization problems and vectorial optimization problems, respectively. Finally, we construct the equivalence relations between the Levitin?CPolyak well-posedness of scalar optimization problems and the vectorial optimization problems. 相似文献
10.
In this article, using the nonlinear scalarization approach by virtue of the nonlinear scalarization function, commonly known as the Gerstewitz function in the theory of vector optimization, Hölder continuity of solution mappings for both set-valued and single-valued cases to parametric vector equilibrium problems is studied. The nonlinear scalarization function is a powerful tool that plays a key role in the proofs, and its main properties (such as sublinearity, continuity, convexity) are fully employed. Especially, its locally and globally Lipschitz properties are provided and the Lipschitz property is first exploited to investigate the Hölder continuity of solutions. 相似文献
11.
Well-Posedness and Scalarization in Vector Optimization 总被引:8,自引:0,他引:8
E. Miglierina E. Molho M. Rocca 《Journal of Optimization Theory and Applications》2005,126(2):391-409
In this paper, we study several existing notions of well- posedness for vector optimization problems. We separate them into two classes and we establish the hierarchical structure of their relationships. Moreover, we relate vector well-posedness and well-posedness of an appropriate scalarization. This approach allows us to show that, under some compactness assumption, quasiconvex problems are well posed.The authors thank Professor C. Zălinescu for pointing out some inaccuracies in Ref. 11. His remarks allowed the authors to improve the present work. 相似文献
12.
In this paper, we consider nonsmooth vector variational-like inequalities and nonsmooth vector optimization problems. By using the scalarization method, we define nonsmooth variational-like inequalities by means of Clarke generalized directional derivative and study their relations with the vector optimizations and the scalarized optimization problems. Some existence results for solutions of our nonsmooth variational-like inequalities are presented under densely pseudomonotonicity or pseudomonotonicity assumption. 相似文献
13.
César Gutiérrez Bienvenido Jiménez Vicente Novo 《Computational Optimization and Applications》2006,35(3):305-324
This work deals with approximate solutions in vector optimization problems. These solutions frequently appear when an iterative
algorithm is used to solve a vector optimization problem. We consider a concept of approximate efficiency introduced by Kutateladze
and widely used in the literature to study this kind of solutions. Necessary and sufficient conditions for Kutateladze’s approximate
solutions are given through scalarization, in such a way that these points are approximate solutions for a scalar optimization
problem. Necessary conditions are obtained by using gauge functionals while monotone functionals are considered to attain
sufficient conditions. Two properties are then introduced to describe the idea of parametric representation of the approximate
efficient set. Finally, through scalarization, characterizations and parametric representations for the set of approximate
solutions in convex and nonconvex vector optimization problems are proved and the obtained results are applied to Pareto problems.
AMS Classification:90C29, 49M37
This research was partially supported by Ministerio de Ciencia y Tecnología (Spain), project BFM2003-02194. 相似文献
14.
Petra Weidner 《Optimization》2018,67(7):1121-1141
Scalarization in vector optimization is often closely connected to the minimization of Gerstewitz functionals. In this paper, the minimizer sets of Gerstewitz functionals are investigated. Conditions are given under which such a set is nonempty and compact. Interdependencies between solutions of problems with different parameters or with different feasible point sets are shown. Consequences for the parameter control in scalarization methods are proved. It is pointed out that the minimization of Gerstewitz functionals is equivalent to an optimization problem which generalizes the scalarization by Pascoletti and Serafini. The results contain statements about minimizers of certain Minkowski functionals and norms. Some existence results for solutions of vector optimization problems are derived. 相似文献
15.
S. Khoshkhabar-amiranloo 《Nonlinear Analysis: Theory, Methods & Applications》2012,75(3):1429-1440
This paper deals with set-valued vector optimization problems and set-valued vector variational inequalities in topological vector spaces, and provides some scalarization approaches for these problems by means of the polar cone and Gerstewitz’s scalarization functions. 相似文献
16.
向量映射的鞍点和Lagrange对偶问题 总被引:4,自引:0,他引:4
本文研究拓扑向量空间广义锥-次类凸映射向量优化问题的鞍点最优性条件和Lagrange对偶问题,建立向量优化问题的Fritz John鞍点和Kuhn-Tucker鞍点的最优性条件及其与向量优化问题的有效解和弱有效解之间的联系。通过对偶问题和向量优化问题的标量化刻画各解之间的关系,给出目标映射是广义锥-次类凸的向量优化问题在其约束映射满足广义Slater约束规格的条件下的对偶定理。 相似文献
17.
In this paper, two kinds of Hadamard well-posedness for vector-valued optimization problems are introduced. By virtue of scalarization
functions, the scalarization theorems of convergence for sequences of vector-valued functions are established. Then, sufficient
conditions of Hadamard well-posedness for vector optimization problems are obtained by using the scalarization theorems. 相似文献
18.
Existence and Optimality Conditions for Approximate Solutions to Vector Optimization Problems 总被引:1,自引:0,他引:1
In this paper, we introduce a new concept of ϵ-efficiency for vector optimization problems. This extends and unifies various notions of approximate solutions in the literature.
Some properties for this new class of approximate solutions are established, and several existence results, as well as nonlinear
scalarizations, are obtained by means of the Ekeland’s variational principle. Moreover, under the assumption of generalized
subconvex functions, we derive the linear scalarization and the Lagrange multiplier rule for approximate solutions based on
the scalarization in Asplund spaces. 相似文献
19.
Duality framework on vector optimization problems in a locally convex topological vector space are established by using scalarization with a cone-strongly increasing function.The dualities for the scalar convex composed optimization problems and for general vector optimization problems are studied.A general approach for studying duality in vector optimization problems is presented. 相似文献
20.
Efficiency and Henig Efficiency for Vector Equilibrium Problems 总被引:6,自引:0,他引:6
We introduce the concept of Henig efficiency for vector equilibrium problems, and extend scalarization results from vector optimization problems to vector equilibrium problems. Using these scalarization results, we discuss the existence of the efficient solutions and the connectedness of the set of Henig efficient solutions to the vector-valued Hartman–Stampacchia variational inequality. 相似文献