Second-Order Conditions for Open-Cone Minimizers and Firm Minimizers in Set-Valued Optimization Subject to Mixed Constraints

Khánh, Phan Quốc; Tung, Nguyen Minh

doi:10.1007/s10957-016-0995-x

Cited by 10 publications

(7 citation statements)

References 33 publications

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“…The above problem subsumes various vector-and set-valued optimization problems while the concept of Q-solution, under a suitable choice of Q, subsumes various kinds of solutions; cf. [42]. It is worth noting the two specific features of the second-order necessary condition we present below: the regularity condition plays an important role and the right-hand side of the multiplier rule (5.6) is not the number 0 as in the classical result (and also in many its developments until now) and it may be strictly negative in particular cases.…”

Section: Alongside the Ordering Cone C We Consider Another Proper Opementioning

confidence: 91%

mentioning

confidence: 99%

“…Hence, a development of our regularity model corresponding to directional metric subregularity is on the agenda. Such an extension is going to properly improve [42,Theorem 3.1].…”

mentioning

confidence: 99%

“…4. Following [42], one can improve Theorem 5.2 by relaxing the restrictive assumption of nonemptyness of the interior of the set ∆ (u,v,k) (IT 2 (S, x, u)).…”

mentioning

confidence: 99%

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An Induction Theorem and Nonlinear Regularity Models

Khánh¹,

Kruger²,

Thao³

2015

SIAM J. Optim.

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A general nonlinear regularity model for a set-valued mapping F : X × R + ⇒ Y , where X and Y are metric spaces, is studied using special iteration procedures, going back to Banach, Schauder, Lyusternik and Graves. Namely, we revise the induction theorem from Khanh, J. Math. Anal. Appl., 118 (1986) and employ it to obtain basic estimates for exploring regularity/openness properties. We also show that it can serve as a substitution of the Ekeland variational principle when establishing other regularity criteria. Then, we apply the induction theorem and the mentioned estimates to establish criteria for both global and local versions of regularity/openness properties for our model and demonstrate how the definitions and criteria translate into the conventional setting of a set-valued mapping F : X ⇒ Y . An application to second-order necessary optimality conditions for a nonsmooth set-valued optimization problem with mixed constraints is provided.

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Section: Alongside the Ordering Cone C We Consider Another Proper Opementioning

confidence: 91%

mentioning

confidence: 99%

“…Hence, a development of our regularity model corresponding to directional metric subregularity is on the agenda. Such an extension is going to properly improve [42,Theorem 3.1].…”

mentioning

confidence: 99%

“…4. Following [42], one can improve Theorem 5.2 by relaxing the restrictive assumption of nonemptyness of the interior of the set ∆ (u,v,k) (IT 2 (S, x, u)).…”

mentioning

confidence: 99%

See 2 more Smart Citations

An Induction Theorem and Nonlinear Regularity Models

Khánh¹,

Kruger²,

Thao³

2015

SIAM J. Optim.

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“…In the vector approach, many types of generalized derivatives were employed to establish the optimality conditions for set-valued optimization problems in two directions: the dual spaces and the primal spaces. For some recent results in this direction, see the books [12,22], the papers [2,3,7,10,13,19,20,24] and the references therein. In the set approach, the optimality conditions for some types of solutions of set optimization problems were established in terms of the directional derivatives of the continuous selections in [1].…”

Section: Introductionmentioning

confidence: 99%