On linear and linearized generalized semi-infinite optimization problems

We consider the local and global topological structure of the feasible set M of a generalized semiinfinite optimization problem. Under the assumption that the defining functions for M are affine-linear with respect to the ind ex variable and separable with respect to the index and the state variable, M can globally be written as the finite union of certain open and closed s ets. Here, it is not necessary to impose any kind of constraint qualificati on on the lower level problem. In fact, these sets are level sets of the lower level Lagrangian, and the o pen sets are generated exactly by Lagrange multiplier vectors with vanishin g entry corresponding to the lower level objective function. This result gi ves rise to a first order necessary optimality condition for the considered generalized semi-infinite problem. Finally it is shown that the description of M by open and closed level sets of the lower level Lagrangian locally carries over to points of the so-cal led mai-type, where neither the linearity nor the separability assumption i s satisfied.