Bilevel Programming: Algorithms

In its generality, bilevel programming constitutes a very difficult class of global optimization problems and can be addressed as such, for instance by customizing metaheuristic procedures such as tabu search ([1]) or genetic algorithms ([5]; cf. also Genetic algorithms). Several research papers on the algorithmic aspects of bilevel programming have actually been published in the Journal of Global Optimization.

Due to the geometric complexity of the constraint set, which is implicitly defined by the lower level program, direct descent methods are difficult to implement. Determining the steepest descent direction, even in the case of linear bilevel programming, is NP-hard ([6]). Whenever the solution set y(x) of the lower level problem is a singleton, the original bilevel program can be rewritten in the form

Feasible descent directions for the function v can be obtained from sensitivity analysis of the lower level problem, and used to construct a generic descent algorithm.

Some solution...