A nonsmooth symmetric reduction ansatz (NSRA) for the smooth generalized semi-infinite programming problem (GSIP) is introduced. It is proved that under NSRA the feasible set of GSIP can locally be described as the feasible set of a so-called disjunctive optimization problem which is defined by finitely many inequality constraints of maximum type. This result in particular shows the appearance of re-entrant corners in the topological closure of the feasible set of GSIP. Forming concepts of a nondegenerate KKT point and a corresponding GSIP-index, the authors next derive a local cell-attachment result for GSIP in the sense of Morse theory. They furthermore prove that NSRA is generic and stable at all KKT points for GSIP and that all such KKT points are nondegenerate. At last the special case of a smooth semi-infinite programming problem (SIP) is considered and the well-known reduction ansatz for SIP is related to NRSA for GSIP.