The paper presents a new method for solving bilevel programming problem by adopting the concepts of an exterior penalty method. The bilevel programming problem is composed of an upper and a lower level. The upper level determines the optimal value of an unknown parameter contained in the lower level optimization problem, based on upper objective and constraint functions, while the lower level problem can be solved under the parameter assigned by upper level. Furthermore the determination of the parameter is performed by considering the optimal solution to the lower level problem corresponding to the parameter. Such a problem cannot be solved by ordinary mathematical programming.
In our exterior penalty approach, augmented objective functions are introduced first, by which the objective function is combined with the constraint function, in the upper and the lower levels, respectively. Then, the original constrained bilevel problem is transformed into an unconstrained bilevel problem, which is solvable by use of a gradient method, and the lower parametric solution is approximated by a differentiable implicit function satisfying the stationary condition for the lower augmented objective function. And also, it can be proved that the solution to the original bilevel problem can be obtained as an accumulation point of a sequence of solutions to the transformed bilevel problem, when the penalty parameters diverge to the infinite in the upper and the lower levels simultaneously. In the points mentioned above, the concepts of our exterior penalty approach is different from an ordinary exterior penalty method.