Bilevel Programming: Implicit Function Approach

If the lower level problem has a unique, strongly stable global optimal solution for all parameter values, then the bilevel optimization problem reduces to a single-level problem. Under certain assumptions, the objective function of this problem is directionally differentiable and locally Lipschitz continuous which can be used to derive necessary and sufficient optimality conditions and to apply algorithms of nonsmooth optimization to solve it. To check satisfaction of the optimality conditions, a bilevel optimization problem with a quadratic optimization problem in the lower level needs to be solved, the bundle algorithm converges to a Clarke stationary point.