We consider a difficult class of optimization problems that we call a
mathematical program with vanishing constraints. Problems of this kind
arise in various applications including optimal topology design problems
of mechanical structures. We show that some standard constraint
qualifications like LICQ and MFCQ usually do not hold at a local minimum
of our program, whereas the Abadie constraint qualification is sometimes
satisfied. We also introduce a suitable modification of the standard
Abadie constraint qualification as well as a corresponding optimality
condition, and show that this modified constraint qualification holds
under fairly mild assumptions. Finally, we discuss the relation between
our class of optimization problems with vanishing constraints and a
mathematical program with equilibrium constraints.

Key Words:
Constrained optimization, vanishing constraints, structural optimization,
constraint qualifications, optimality conditions, mathematical programs
with equilibrium constraints.