One-parameter families of feasible sets in semi-infinite optimization

Feasible sets in semi-infinite optimization are basically defined by means of infinitely many inequality constraints. We consider one-parameter famili es of such sets. In particular, all defining functions - including those de fining the index set of the inequality constraints - will depend on a param eter. We note that a semi-infinite problem is a two-level problem in the se nse that a point is feasible if and only if all global minimizers of a corr esponding marginal function are nonnegative. For a quite natural class of mappings we characterize changes in the global topological structure of the corresponding feasible set as the parameter v aries. As long as the index set (-mapping) of the inequality constraints is lower semicontinuous, all changes in topology are those which generically appear in one-parameter sets defined by finitely many constraints. In the c ase, however, that some component of the mentioned index set is born (or va nishes), the topological change is of global nature and is not controllable . In fact, the change might be as drastic as that when adding or deleting a n (arbitrary) inequality constraint.