Mathematical programs with vanishing constraints: optimality conditions and constraint qualifications

2007 ◽  
Vol 114 (1) ◽  
pp. 69-99 ◽  
Author(s):  
Wolfgang Achtziger ◽  
Christian Kanzow
Keyword(s):  
Optimality Conditions ◽  
Constraint Qualifications ◽  
Mathematical Programs ◽  
Vanishing Constraints

scholarly journals On an exact penality result and new constraint qualifications for mathematical programs with vanishing constraints

2019 ◽  
Vol 29 (3) ◽  
pp. 309-324
Author(s):  
Triloki Nath ◽  
Abeka Khare
Keyword(s):  
Penalty Function ◽  
Constraint Qualification ◽  
Constraint Qualifications ◽  
Mathematical Programs ◽  
Vanishing Constraints ◽  
Abadie Constraint Qualification

In this paper, we considered the mathematical programs with vanishing constraints or MPVC. We proved that an MPVC-tailored penalty function, introduced in [5], is still exact under a very weak and new constraint qualification. Most importantly, this constraint qualification is shown to be strictly stronger than MPVC-Abadie constraint qualification.

scholarly journals Optimality Conditions, Qualifications and Approximation Method for a Class of Non-Lipschitz Mathematical Programs with Switching Constraints

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2915
Author(s):  
Jinman Lv ◽  
Zhenhua Peng ◽  
Zhongping Wan
Keyword(s):  
Optimality Conditions ◽  
Approximation Method ◽  
Constraint Qualifications ◽  
Necessary Optimality Conditions ◽  
Accumulation Point ◽  
Local Minimizer ◽  
Necessary Condition ◽  
Local Minimizers ◽  
Mathematical Programs ◽  
The Given

In this paper, we consider a class of mathematical programs with switching constraints (MPSCs) where the objective involves a non-Lipschitz term. Due to the non-Lipschitz continuity of the objective function, the existing constraint qualifications for local Lipschitz MPSCs are invalid to ensure that necessary conditions hold at the local minimizer. Therefore, we propose some MPSC-tailored qualifications which are related to the constraints and the non-Lipschitz term to ensure that local minimizers satisfy the necessary optimality conditions. Moreover, we study the weak, Mordukhovich, Bouligand, strongly (W-, M-, B-, S-) stationay, analyze what qualifications making local minimizers satisfy the (M-, B-, S-) stationay, and discuss the relationship between the given MPSC-tailored qualifications. Finally, an approximation method for solving the non-Lipschitz MPSCs is given, and we show that the accumulation point of the sequence generated by the approximation method satisfies S-stationary under the second-order necessary condition and MPSC Mangasarian-Fromovitz (MF) qualification.

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