Stability in disjunctive linear optimization I: Continuity of the feasible set

The aim of this paper is to study the continuous dependence of the feasible set of a disjunctive semi-infinite linear optimization problem on all involved parameters (matrix and right-hand side). The feasible set of such an optimization problem is the union of (a possible infinite number of) convex sets, which each is described by a finite or an infinite number of strict and non-strict linear inequalities. The paper derives necessary and sufficient conditions for the upper- and lower-semi-continuity, and the closedness of the feasible-set-mapping Z. Especially, the compactness of the boundary of the feasible set and the closedness of Z are equivalent to the upper-semi-continuity of Z, while the lower-semi-continuity of Z is equivalent to a certain constraint qualification. This constraint qualification is a strengthened kind of Slater condition. From these investigations, the paper derives known results in parametric semi-infinite optimization and parametric integer programming.