A general vectorial Ekeland’s variational principle with a P-distance

In this paper, we characterize the nonemptiness of the set of weak minimal elements for a nonempty subset of a linear space. Moreover, we obtain some existence results for a nonconvex set-valued optimization problem under weaker topological conditions. Introduction and preliminariesLet Y be a real linear space ordered by a convex cone C ⊆ Y which is assumed to be proper; i.e., {0} = C = Y . Let K be a nonempty set and F : K ⇒ Y be a set-valued mapping with nonempty values. A general form of set-valued optimization problem is usually defined as follows:(SOP) min F(x) subject to x ∈ K .There are two approaches to defining the solutions of this problem: the vector approach [3,14,17] and the set approach [15,16]. In the vector approach,x ∈ K is a solution of the problem (SOP), whenever F(x) contains a weak minimal element or minimal element of F(K ) = ∪ x∈K F(x). In the set approach, it is necessary to introduce an ordering for sets and find a minimal element of subset {F(x) : x ∈ K } of P(Y ), where P(Y ) is the set of B M. Fakhar

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“…As a consequence of the above theorem we obtain the following result due to Qui and He [19,Proposition 2.3].…”

Section: Some Properties Of Vectorially Closed Setsmentioning

confidence: 75%

“…For more details on the following concepts and for some examples of these notions we can refer to [1,[13][14][15]19].…”

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confidence: 99%

Weak minimal elements and weak minimal solutions of a nonconvex set-valued optimization problem

Chinaie

Fakhar

et al. 2019

J Glob Optim

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“…For subsets of real linear spaces, we recall some concepts of vectorial closedness (see, e.g., [1,17,27]). Let A be a nonempty subset of a real linear space Y .…”

Section: Definition 22mentioning

confidence: 99%

“…In this paper, we try to give a general vectorial Ekeland's variational principle, where the objective function is defined on an F-type topological space and taking values in a pre-ordered linear space (or a pre-ordered locally convex space). Being quite different from the previous versions of vectorial Ekeland's variational principle, in our version the perturbation is no longer dependent on a fixed positive vector k 0 (see, e.g., [5,6,12,13,15,20,25,[27][28][29]) or a fixed family {k α } α of positive vectors (see [30]). It contains a family of set-valued functions taking values in the positive cone K and a family of subadditive functions of topology generating quasi-metrics.…”

Section: Introductionmentioning

confidence: 99%

Vectorial variational principle with variable set-valued perturbation

Zhang

Qiu

2015

Acta. Math. Sin.-English Ser.

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We give a general vectorial Ekeland's variational principle, where the objective function is defined on an F-type topological space and taking values in a pre-ordered real linear space. Being quite different from the previous versions of vectorial Ekeland's variational principle, the perturbation in our version is no longer only dependent on a fixed positive vector or a fixed family of positive vectors. It contains a family of set-valued functions taking values in the positive cone and a family of subadditive functions of topology generating quasi-metrics. Hence, the direction of the perturbation in the new version is a family of variable subsets which are dependent on the objective function values. The general version includes and improves a number of known versions of vectorial Ekeland's variational principle. From the general Ekeland's principle, we deduce the corresponding versions of Caristi-Kirk's fixed point theorem and Takahashi's nonconvex minimization theorem. Finally, we prove that all the three theorems are equivalent to each other.

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“…The following results concerning Gerstewitz's functions originate from [18,19]. [19,43,44]). Let D ⊂ Y be a convex cone and k 0 ∈ D\ − vcl(D).…”

Section: Generalized Gerstewitz's Functions and Their Propertiesmentioning

confidence: 99%

Generalized Gerstewitz’s Functions and Vector Variational Principle for ϵ-Efficient Solutions in the Sense of Németh

Qiu

2018

Acta. Math. Sin.-English Ser.

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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 ǫ-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.

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A general vectorial Ekeland’s variational principle with a P-distance

Cited by 36 publications

References 39 publications

Weak minimal elements and weak minimal solutions of a nonconvex set-valued optimization problem

Weak minimal elements and weak minimal solutions of a nonconvex set-valued optimization problem

Vectorial variational principle with variable set-valued perturbation

Generalized Gerstewitz’s Functions and Vector Variational Principle for ϵ-Efficient Solutions in the Sense of Németh

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