Nested Optimization of Hierarchical Systems

What does the decision-making involve in hierarchical organizations? How to conciliate the overall objectives at the top level of a firm, with the productivity and marketing objectives at a lower level? A hierarchy of decision-makers is a typical situation for local governments and planning agency. The decision variables are partitioned among an upper level and different lower levels. The programming problem looks like a set of nested programming problems with agents belonging to hierarchical levels. The problem is similar to a static noncooperative two-person game by Stackelberg. Within each level, the agents play a -person non-zero-sum game. Between the levels, the agents play a -person Stackelberg game. Let the problem correspond to a bilevel programming (BLP) problem. The two players optimize their payoffs by controlling their decision variables. Both players have perfect information about the objectives and strategies of the opponent. The leader plays first but must anticipate all the possible reactions, and the followers react optimally.The algorithm for finding the Nash- Stackelberg solution belongs to the four classes of solution methods. The methods are a reformulation by using optimality conditions, the penalty method, and metaheuristics such as with SA or GAs algorithms. Thus, under convexity and regularity conditions, the initial problem can be reformulated as a single nonlinear optimization by using KKT optimality conditions. The -best algorithm computes the global solution of BLP by enumerating the extreme points of the constrained region. Several problems illustrate the process, such as Bard’s BLP problem, and two other challenges with one and two followers.

Keywords: Bilevel programming, Follower, Hierarchical optimization, Inductible region, KKT optimality conditions, Kth-best method, Leader, Lower level programming, Multi-agent, Multilevel programming, Nash equilibrium, Nash- Stackelberg solution, Nested optimization, Noncooperative two-person game, Parametric complementing pivot, Rational reaction set, Reverse convex program, Stackelberg game, Upper-level programming.