Stationary conditions for mathematical programs with vanishing constraints using weak constraint qualifications.

*(English)*Zbl 1141.90572Summary: We consider a class of optimization problems that is called a mathematical program with vanishing constraints (MPVC for short). This class has some similarities to mathematical programs with equilibrium constraints (MPECs for short), and typically violates standard constraint qualifications, hence the well-known Karush-Kuhn-Tucker conditions do not provide necessary optimality criteria. In order to obtain reasonable first order conditions under very weak assumptions, we introduce several MPVC-tailored constraint qualifications, discuss their relation, and prove an optimality condition which may be viewed as the counterpart of what is called M-stationarity in the MPEC-field.

##### MSC:

90C46 | Optimality conditions and duality in mathematical programming |

##### Keywords:

Mathematical programs with vanishing constraints; mathematical programs with equilibrium constraints; optimality conditions; constraint qualifications; limiting normal cone##### Software:

MacMPEC
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\textit{T. Hoheisel} and \textit{C. Kanzow}, J. Math. Anal. Appl. 337, No. 1, 292--310 (2008; Zbl 1141.90572)

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