New Second-Order Tangent Epiderivatives and Applications to Set-Valued Optimization

Peng, Zhenhua; Xu, Yi

doi:10.1007/s10957-016-1011-1

Cited by 10 publications

(5 citation statements)

References 18 publications

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“…Similar to the proof of Lemma 3.3 in [9], we can get the following Lemmas 2.16 and 2.17. Lemma 2.16.…”

Section: Remark 2 ([9]mentioning

confidence: 66%

“…Lemma 2.15. ( [9])Let S be a nonempty convex subset of X and G : X → 2 Z be a set-valued map. Suppose that G is a D-function and lower semicontinuous on…”

Section: Remark 2 ([9]mentioning

confidence: 99%

“…(i) Linear condition is reduced to nonlinear condition; (ii) A variable ordering structure is considered in Corollary 2, a fixed ordering structure is considered in Theorem 4.3 of [9]. A variable ordering structure is a generalization of a fixed ordering structure.…”

Section: Remark 2 ([9]mentioning

confidence: 99%

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Second-Order characterizations for set-valued equilibrium problems with variable ordering structures

Hu¹,

Xu²,

Zhang³

2022

JIMO

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<p style='text-indent:20px;'>The concepts of weakly efficient solutions and globally efficient solutions are introduced for constrained set-valued equilibrium problems with variable ordering structures. By applying the second-order tangent epiderivative and a nonlinear functional, necessary optimality conditions for weakly efficient solutions and globally efficient solutions are established without any convexity assumption. Under the cone-convexity of the objective and constraint functions, sufficient optimality conditions are given. In addition, the tangent derivatives of objective and constraint functions are separated. Simultaneously, a unified necessary and sufficient optimality conditions for weakly efficient solutions is derived, and the same goes for globally efficient solutions. In particular, we give specific examples to illustrate the optimality conditions, respectively.</p>

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“…Similar to the proof of Lemma 3.3 in [9], we can get the following Lemmas 2.16 and 2.17. Lemma 2.16.…”

Section: Remark 2 ([9]mentioning

confidence: 66%

“…Lemma 2.15. ( [9])Let S be a nonempty convex subset of X and G : X → 2 Z be a set-valued map. Suppose that G is a D-function and lower semicontinuous on…”

Section: Remark 2 ([9]mentioning

confidence: 99%

See 1 more Smart Citation

Second-Order characterizations for set-valued equilibrium problems with variable ordering structures

Hu¹,

Xu²,

Zhang³

2022

JIMO

Self Cite

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show abstract

“…Similarly to Definition 4.3 in [18], we introduce the following second-order generalized lower (upper) directional derivative for vector-valued functions.…”

Section: Second-order Necessary Optimality Conditionsmentioning

confidence: 99%

“…For example, Gong et al [1] introduced the concept of radial tangent cone and presented several kinds of necessary and sufficient conditions for some proper efficiencies, Kasimbeyli [2] gave necessary and sufficient optimality conditions based on the concept of the radial epiderivatives. At the same time, second-order optimality conditions and higher-order optimality conditions for vector optimization problems have been extensively studied in the literature (see [3–18]). Jahn et al [3] proposed second-order epiderivatives for set-valued maps, and by using these concepts they gave second-order necessary optimality conditions and a sufficient optimality condition in set optimization.…”

Section: Introductionmentioning

confidence: 99%

Second-order lower radial tangent derivatives and applications to set-valued optimization

Peng

2017

J Inequal Appl

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We introduce the concepts of second-order radial composed tangent derivative, second-order radial tangent derivative, second-order lower radial composed tangent derivative, and second-order lower radial tangent derivative for set-valued maps by means of a radial tangent cone, second-order radial tangent set, lower radial tangent cone, and second-order lower radial tangent set, respectively. Some properties of second-order tangent derivatives are discussed, using which second-order necessary optimality conditions are established for a point pair to be a Henig efficient element of a set-valued optimization problem, and in the expressions the second-order tangent derivatives of the objective function and the constraint function are separated.

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Second-Order Karush–Kuhn–Tucker Optimality Conditions for Set-Valued Optimization Subject to Mixed Constraints

Peng

Wan

2018

Results Math

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New Second-Order Tangent Epiderivatives and Applications to Set-Valued Optimization

Cited by 10 publications

References 18 publications

Second-Order characterizations for set-valued equilibrium problems with variable ordering structures

Second-Order characterizations for set-valued equilibrium problems with variable ordering structures

Second-order lower radial tangent derivatives and applications to set-valued optimization

Second-Order Karush–Kuhn–Tucker Optimality Conditions for Set-Valued Optimization Subject to Mixed Constraints

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