Trap-doors in the solution set of semi-infinite optimization problems

Abstract. For one-parametric finite optimization problems it is well-known that, in the generic case, each generalized critical point belongs to one of exactly five different types, where the degenerate points (type 2 to 5) form the boundary of the set of nondegenerate points (type 1). In particular, the set of g.c.points is closed. This is not the case in generic one-parametric semi-infinite programming, where three additional types of degenerate g.c.points occur. In fact, we present an example which shows that the endpoint of a path of nondegenerate critical points does not have to be a g.c.point. Moreover, a pathfollowing algorithm cannot detect this point by using only local information, and even if global tests allow a detection, no jump direction can be given. We discuss the consequences of this phenomenon for pathfollowing methods in semi-infinite programming. Under the additional assumption of continuous dependence of the index set on the parameter, we derive the closedness of the g.c.point set.