A new computational method for Stackelberg and min-max problems by use of a penalty method

This paper is concerned with the Stackelberg problem and the min-max problem in competitive systems. The Stackelberg approach is applied to the optimization of two-level systems where the higher level determines the optimal value of its decision variables (parameters for the lower level) so as to minimize its objective, while the lower level minimizes its own objective with respect to the lower level decision variables under the given parameters. Meanwhile, the min-max problem is to determine a min-max solution such that a function maximized with respect to the maximizer's variables is minimized with respect to the minimizer's variables. This problem is also characterized by a parametric approach in a two-level scheme. New computational methods are proposed here; that is, a series of nonlinear programming problems approximating the original two-level problem by application of a penalty method to a constrained parametric problem in the lower level are solved iteratively. It is proved that a sequence of approximated solutions converges to the correct Stackelberg solution, or the min-max solution. Some numerical examples are presented to illustrate the algorithms.