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.