Decomposition methods for parameter identification and bilevel programming

Abstract

This thesis deals with optimization problems that involve complex subtasks. Treating these problems as nonlinear programs often leads to difficulties in the numerical solution, for example the convergence to undesired local solutions for poor initial guesses. To circumvent these problems, the concept of decomposition is introduced. The complex subtask is modeled by artificial variables and constraints. On the one hand, this increases the dimensionality of the problem, which on the other hand allows a user to employ suitable initialization strategies. This idea is applied to the problem classes of nonlinear parameter identification for dynamical systems and bilevel optimization. For the latter, a reformulation method is developed, which is based on embedding a fixed number of Sequential Quadratic Programming iteration steps to solve the subtask given as a nonlinear program. The focus of this work is on the numerical verification of different decomposition approaches. Using the examples of a pendulum and a robotic system, it is demonstrated that the choice of the problem formulation in combination with a suitable initialization strategy influences the region of attraction of the global solution of the associated parameter identification problem. Finally, an extension of a single shooting homotopy approach is presented. In the area of bilevel optimization, examples are used to show that the number of embedded iteration steps increases the region of attraction of the global solution. The newly developed method is also compared with other established methods. Using a collection of bilevel problems, its handling, flexibility, and efficiency are investigated. It is also demonstrated for both problem classes that a decomposition has only a minor impact on computation times due to the exploitation of sparsity. In addition, decomposition allows to keep the required number of iterations stable despite poor initial guesses. The obtained results show that both problem classes can be considered under a common aspect. Important criteria, such as the robustness or the region of attraction of the global solution, can be influenced solely by reformulating the problem.