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 index 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 sets. Here, it is not necessary to impose any kind of constraint qualification on the lower level problem. In fact, these sets are level sets of the lower level Lagrangian, and the open sets are generated exactly by Lagrange multiplier vectors with vanishing entry corresponding to the lower level objective function. This result gives 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-called mai-type, where neither the linearity nor the separability assumption is satisfied.