An investigation of the linear three level programming problem

The open-loop Stackelberg game is conceptually extended to p players by the multilevel programming problem (MLPP) and can thus be used as a model for a variety of hierarchical systems in which sequential planning is the norm. In this paper the rational reaction sets for each of the players is first developed, and then the geometric properties of the linear MLPP are stated. Next, first-order necessary conditions are derived and the problem is recast as a standard nonlinear program. A cutting plane algorithm employing a vertex search procedure at each iteration is proposed to solve the linear three-level case. An example is given to highlight the results along with some computational experience. Although only a limited number of test problems were examined, the amount of work required to find a solution seems to grow with the square of the row dimensions of the underlying polyhedral constraint set.