A novel approach to Bilevel nonlinear programming

Recently developed methods of monotonic optimization have been applied successfully for studying a wide class of nonconvex optimization problems, that includes, among others, generalized polynomial programming, generalized multiplicative and fractional programming, discrete programming, optimization over the efficient set, complementarity problems, etc. In the present paper the monotonic approach is extended to the following General Bilevel Programming Problem min F (x, y) s.t. g1(x, y) ≤ 0, x ∈ R1 + y solves min{d(y)| g2(C(x), y) ≤ 0, y ∈ R2 + }    (GBP) where g1(x, y) : R1 + ×R2 + → R, g2(u, y) : Rm×R2 + → R, d : R2 + → R, are continuous functions, C : R1 → R is a continuous mapping, and for every fixed y, g2(u, y) is a decreasing function of u. It is shown that (GBP) can be transformed into a monotonic optimization problem in R, which can be solved by “polyblock” approximation or, more efficiently, by a branch-reduce-and-bound method using monotonicity cuts. The method is particularly suitable for Bilevel Convex Programming (case when all functions involved are convex) and Bilevel Linear Programming .