Solving Bilevel Optimization Problems Using Kriging Approximations

Bilevel optimization involves two levels of optimization, where one optimization problem is nested within the other. The structure of the problem often requires solving a large number of inner optimization problems that make these kinds of optimization problems expensive to solve. The reaction set mapping and the lower level optimal value function mapping are often used to reduce bilevel optimization problems to a single level; however, the mappings are not known a priori, and the need is to be estimated. Though there exist a few studies that rely on the estimation of these mappings, they are often applied to problems where one of these mappings has a known form, that is, piecewise linear, convex, etc. In this article, we utilize both these mappings together to solve general bilevel optimization problems without any assumptions on the structure of these mappings. Kriging approximations are created during the generations of an evolutionary algorithm, where the population members serve as the samples for creating the approximations. One of the important features of the proposed algorithm is the creation of an auxiliary optimization problem using the Kriging-based metamodel of the lower level optimal value function that solves an approximate relaxation of the bilevel optimization problem. The auxiliary problem when used for local search is able to accelerate the evolutionary algorithm toward the bilevel optimal solution. We perform experiments on two sets of test problems and a problem from the domain of control theory. Our experiments suggest that the approach is quite promising and can lead to substantial savings when solving bilevel optimization problems. The approach is able to outperform state-of-the-art methods that are available for solving bilevel problems, in particular, the savings in function evaluations for the lower level problem are substantial with the proposed approach.