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Mathematical programs with vanishing constraints: optimality conditions and constraint qualifications. (English) Zbl 1151.90046
In this very interesting paper, a new class (MPVC) of optimization problems is considered where a part of the constraints needs to be satisfied only if the variables are contained in some specified set and are absent otherwise. Besides interesting applications, the relations to mathematical programs with equilibrium constraints (MPEC) are explained. Regularity conditions like LICQ or MFCQ are violated at any feasible point of an MPEC, for MPVC the situation is better. The mentioned as well as the Abadie constraint qualification can be satisfied under certain assumptions which is investigated in detail in the paper. If the Abadie constraint qualification (which is weaker than MFCQ) is satisfied, then the Karush-Kuhn-Tucker necessary optimality conditions are satisfied at a local optimum of MPVC. In the last part of the paper a modified Abadie constraint qualification tailored to the special structure of the problem is introduced and its usefulness for deriving optimality conditions is shown.

MSC:
90C30 Nonlinear programming
90C46 Optimality conditions and duality in mathematical programming
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