arXiv subjects

Jing-Hui Qiu

Publications and source records attributed to Jing-Hui Qiu.

5 recordsLinked to original sources

arXiv2017

Ekeland variational principles with set-valued objective functions and set-valued perturbations

In the setting of real vector spaces, we establish a general set-valued Ekeland variational principle (briefly, denoted by EVP), where the objective function is a set-valued map taking values in a real vector space quasi-ordered by a convex cone $K$ and the perturbation consists of a $K$-convex subset $H$ of the ordering cone $K$ multiplied by the distance function. Here, the assumption on lower boundedness of the objective function is taken to be the weakest kind. From the general set-valued EVP, we deduce a number of particular versions of set-valued EVP, which extend and improve the related results in the literature. In particular, we give several EVPs for approximately efficient solutions in set-valued optimization, where a usual assumption for $K$-boundedness (by scalarization) of the objective function's range is removed. Moreover, still under the weakest lower boundedness condition, we present a set-valued EVP, where the objective function is a set-valued map taking values in a quasi-ordered topological vector space and the perturbation consists of a $σ$-convex subset of the ordering cone multiplied by the distance function.

math.FA↗

arXiv2016

A partial order principle and vector variational principle for $ε$-efficient solutions in the sense of Németh

In this paper, we establish a partial order principle, which is useful to deriving vector Ekeland variational principle (denoted by EVP). By using the partial order principle and extending Gerstewitz's functions, we obtain a vector EVP for $ε$-efficient solutions in the sense of Németh, which essentially improves the earlier results by removing a usual assumption for boundedness of range of the objective function. From this, we also deduce several special vector EVPs, which improve and generalize the related known results.

math.FA↗

arXiv2014

A revised pre-order principle and set-valued Ekeland variational principle

In my former paper "A pre-order principle and set-valued Ekeland variational principle" (see: arXiv: 1311.4951[math.FA]), we established a general pre-order principle. From the pre-order principle, we deduced most of the known set-valued Ekeland variational principles (denoted by EVPs) and their improvements. But the pre-order principle could not imply Khanh and Quy's EVP in [On generalized Ekeland's variational principle and equivalent formulations for set-valued mappings, J. Glob. Optim., 49 (2011), 381-396], where the perturbation contains a weak $τ$-function. In this paper, we give a revised version of the pre-order principle. This revised version not only implies the original pre-order principle, but also can be applied to obtain the above Khanh and Quy's EVP. Thus, the revised pre-order principle implies all the known set-valued EVPs in set containing forms (to my knowledge).

math.FA↗

arXiv2013

Non-Conflicting Ordering Cones and Vector Optimization in Inductive Limits

Let $(E,ξ)={\rm ind}(E_n, ξ_n)$ be an inductive limit of a sequence $(E_n, ξ_n)_{n\in N}$ of locally convex spaces and let every step $(E_n, ξ_n)$ be endowed with a partial order by a pointed convex (solid) cone $S_n$. In the framework of inductive limits of partially ordered locally convex spaces, the notions of lastingly efficient points, lastingly weakly efficient points and lastingly globally properly efficient points are introduced. For several ordering cones, the notion of non-conflict is introduced. Under the requirement that the sequence $(S_n)_{n\in N}$ of ordering cones is non-conflicting, an existence theorem on lastingly weakly efficient points is presented. From this, an existence theorem on lastingly globally properly efficient points is deduced.

math.FA↗

arXiv2013

A pre-order principle and set-valued Ekeland variational principle

We establish a pre-order principle. From the principle, we obtain a very general set-valued Ekeland variational principle, where the objective function is a set-valued map taking values in a quasi ordered linear space and the perturbation contains a family of set-valued maps satisfying certain property. From this general set-valued Ekeland variational principle, we deduce a number of particular versions of set-valued Ekeland variational principle, which include many known Ekeland variational principles, their improvements and some new results.

math.FA↗