Abstract
The Stackelberg problem can be formulated as a bilevel optimization problem. In general, such a problem is not a convex program even if all objective and constraint functions are convex. So it is difficult to obtain the global optimal solution.
In this paper, the Stackelberg problem is equivalently transformed into an optimization problem with an equality and several inequality constraints by introducing the minimal-value function of the lower-level program. Under appropriate assumptions, this problem is again transformed into a program in which the objective and constraints are convex functions. Then, by using the concept of exterior penalty method, we obtain an auxiliary problem having only inequality constraints for the transformed problem. It is proved that the global optimal solution to the transformed problem can be calculated as an accumulation point of a sequence of solutions to the auxiliary problems. After that, we show that the auxiliary problem can be equivalently transformed into a concave program for which we can easily find a global optimal solution.
The proposed method can be applied to a wide class of the Stackelberg problems, in which each function in the upper-level program is convex or defference of two convex functions, and one in the lower-level is convex.
