In the paper, we propose a bilevel direct search method for solving a type of leader-follower problems with each decision maker's objective being a "black-box" function. First, we give a description for a leader-follower optimization problem. Then, we investigate a bilevel direct search method including two algorithms for combinatorially solving the upper and lower level problems, and explore the convergence properties of this bilevel method. Finally, we apply our method to a health insurance problem and carry out several numerical examples to illustrate how the method performs when solving leader-follower problems. In the numerical experiments, we broaden the implementation of the bilevel method in which the direct search algorithm for the lower level problem is replaced by a trust region algorithm.
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