Abstract
This paper presents a new method for solving two-level optimization problems by adopting the concepts of an interior penalty method. The two-level optimization problem is outlined as follows: The two-level optimization 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 which takes the form of a usual optimization problem, based on upper objective and constraint functions, while the lower level problem can be solved under the parameter assigned by the upper level. Furthermore the determination of the parameter is performed by considering of the optimal solution to the lower level corresponding to the parameter. The problem mentioned above cannot be solved by ordinary mathematical programming techniques.
In this paper, 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. Through this approach, the constrained two-level problem is trans-formed into an unconstrained two-level problem, which is solvable by means of unconstrained optimization techniques. It can then be proved that the solution to the original two-level problem can be obtained as an accumulation point of a sequence of solutions to the transformed problems, when the penalty parameters are updated in the upper and lower levels simultaneously. This new method is more efficient and applicable than the other penalty approach proposed previously in Ref. 9).
