Topological Aspects of Nonsmooth Optimization

Shikhman, Vladimir

doi:10.1007/978-1-4614-1897-9

Cited by 17 publications

(16 citation statements)

References 64 publications

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“…Izmailov [14] studied some kind of semi-continuity and Lipshitz continuity of locally optimal solution mapping under some mild conditions. More recently, Jongen et al [16] and Shikhman [30] studied the stability for C-stationarity and S-stationarity and showed that, under suitable conditions, the C-/S-stationary points are strongly stable in the sense of Kojima [18]. A natural question is "are there any stability results for the M-stationarity?…”

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confidence: 99%

Stability Analysis for Parametric Mathematical Programs with Geometric Constraints and Its Applications

Guo¹,

Lin²,

Ye³

2012

SIAM J. Optim.

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This paper studies stability for parametric mathematical programs with geometric constraints. We show that, under the no nonzero abnormal multiplier constraint qualification and the second-order growth condition or second-order sufficient condition, the locally optimal solution mapping and stationary point mapping are nonempty-valued and continuous with respect to the perturbation parameter and, under some suitable conditions, the stationary pair mapping is calm. Furthermore, we apply the above results to parametric mathematical programs with equilibrium constraints. In particular, we show that the M-stationary pair mapping is calm with respect to the perturbation parameter if the M-multiplier second-order sufficient condition is satisfied and the S-stationary pair mapping is calm if the S-multiplier second-order sufficient condition is satisfied and the bidegenerate index set is empty.

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confidence: 99%

Stability Analysis for Parametric Mathematical Programs with Geometric Constraints and Its Applications

Guo¹,

Lin²,

Ye³

2012

SIAM J. Optim.

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“…Now, we recall the definitions of C-MFCQ and C-stationarity. Note that C-MFCQ is called MFC in [10,11,17,18,34].…”

Section: Remark 31mentioning

confidence: 99%

“…Note that the set Tx (r ,s,ρ,σ ) is a so-called tangent space, see e.g. [34]. The next result is obvious and therefore its proof is skipped.…”

Section: A Necessary Condition For Strong Stabilitymentioning

confidence: 99%

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Strongly stable C-stationary points for mathematical programs with complementarity constraints

Escobar

Rückmann

2020

Math. Program.

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In this paper we consider the class of mathematical programs with complementarity constraints (MPCC). Under an appropriate constraint qualification of Mangasarian-Fromovitz type we present a topological and an equivalent algebraic characterization of a strongly stable C-stationary point for MPCC. Strong stability refers to the local uniqueness, existence and continuous dependence of a solution for each sufficiently small perturbed problem where perturbations up to second order are allowed. This concept of strong stability was originally introduced by Kojima for standard nonlinear optimization; here, its generalization to MPCC demands a sophisticated technique which takes the disjunctive properties of the solution set of MPCC into account.

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“…There are many applications where MPCC is used, we refer e.g. to [23,28]. The results of this paper refer to the particular structure imposed by the complementarity constraints.…”

Section: Introductionmentioning

confidence: 99%

MPCC: Strong Stability of M-stationary Points

Günzel

Escobar

Rückmann

2021

Set-Valued Var. Anal

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In this paper we study the class of mathematical programs with complementarity constraints MPCC. Under the Linear Independence constraint qualification MPCC-LICQ we state a topological as well as an equivalent algebraic characterization for the strong stability (in the sense of Kojima) of an M-stationary point for MPCC. By allowing perturbations of the describing functions up to second order, the concept of strong stability refers here to the local existence and uniqueness of an M-stationary point for any sufficiently small perturbed problem where this unique solution depends continuously on the perturbation. Finally, some relations to S- and C-stationarity are briefly discussed.

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Topological Aspects of Nonsmooth Optimization

Cited by 17 publications

References 64 publications

Stability Analysis for Parametric Mathematical Programs with Geometric Constraints and Its Applications

Stability Analysis for Parametric Mathematical Programs with Geometric Constraints and Its Applications

Strongly stable C-stationary points for mathematical programs with complementarity constraints

MPCC: Strong Stability of M-stationary Points

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