On Non-Standard Problem Formulations in Structural Optimization

1. Abstract The talk 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 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, 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 so-called Mathematical Program with Equilibrium Constraints (MPEC). This class of optimization problems has been very well studied by the math programming community 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 on the reformulated problem. 2.