Mathematical programs with a two-dimensional reverse convex constraint

We consider the problem min{f(x) : x E G, T(x) tI. int D}, where fis a lower semicontinuous function, G a compact, nonempty set in JRn, D a closed convex set in JR2 with nonempty interior, and T a continuous mapping from JRn to JR2. The constraint T( x) tI. int D is areverse 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 JR2, and hence can be solved effectively provided f, T are convex and G is convex or discrete. In particular, we discuss areverse convex constraint of the form (c, x) . (d, x) ::; 1. We also compare the approach in this paper with the parametric approach.