Mathematical programs with a two-dimensional reverse convex constraint

We consider the problem min {f(x): x G, T(x) int D}, where f is a lower semicontinuous function, G a compact, nonempty set in ⁿ, D a closed convex set in ² with nonempty interior and T a continuous mapping from ⁿ to ². The constraint T(x) int D is a reverse convex constraint, so the feasible domain may be disconnected even when f, T are affine and G is a polytope. We show that this problem can be reduced to a quasiconcave minimization problem over a compact convex set in ² and hence can be solved effectively provided f, T are convex and G is convex or discrete. In particular we discuss a reverse convex constraint of the form c, x · d, x1. We also compare the approach in this paper with the parametric approach.