A Branch and Bound Algorithm for the Bilevel Programming Problem

The bilevel programming problem is a static Stackelberg game in which two players try to maximize their individual objective functions. Play is sequential and uncooperative in nature. This paper presents an algorithm for solving the linear/quadratic case. In order to make the problem more manageable, it is reformulated as a standard mathematical program by exploiting the follower’s Kuhn–Tucker conditions. A branch and bound scheme suggested by Fortuny-Amat and McCarl is used to enforce the underlying complementary slackness conditions. An example is presented to illustrate the computations, and results are reported for a wide range of problems containing up to 60 leader variables, 40 follower variables, and 40 constraints. The main contributions of the paper are in the step-by-step details of the implementation, and in the scope of the testing.