The paper deals with optimization problems with vanishing constraints, i.e.,
with constraints that must not be considered at certain points of the feasible
domain. Such formulations are typical in Topology Optimization of mechanical
structures where ``mechanical response`` of vanishing structural members gets
lost during optimization, and thus certain function values become undefined.
As an example, consider topology problems with stress constraints. Problem
formulations with vanishing constraints are not covered by the standard
framework of Nonlinear Optimization. As a consequence, standard optimization
theory cannot be applied, standard solution methods generally fail, and
convergence to non-optimal points is observed. Closely related is the effect
of so-called singular optimizers.
The paper deals with reformulations in standard form and with related optimality
conditions. More closely we will focus on a new framework working with a
reformulation in standard form given as a Mathematical Program with Equilibrium
Constraints (MPEC). This class of optimization problems has been very well
studied by the community of Mathematical Programming in the recent past. In
particular, several concepts of optimality conditions have been derived for MPECs.
Surprisingly it turns out that in our particular situation all these concepts
are (practically) equivalent. By this, known numerical procedures for the solution
of MPECs can be applied for the treatment of the reformulated problem.