Levenberg–Marquardt method and partial exact penalty parameter selection in bilevel optimization

Abstract

We consider the optimistic bilevel optimization problem, known to have a wide range of applications in engineering, that we transform into a single-level optimization problem by means of the lower-level optimal value function reformulation. Subsequently, based on the partial calmness concept, we build an equation system, which is parameterized by the corresponding partial exact penalization parameter. We then design and analyze a Levenberg–Marquardt method to solve this parametric system of equations. Considering the fact that the selection of the partial exact penalization parameter is a critical issue when numerically solving a bilevel optimization problem by means of the value function reformulation, we conduct a careful experimental study to this effect, in the context of the Levenberg–Marquardt method, while using the Bilevel Optimization LIBrary (BOLIB) series of test problems. This study enables the construction of some safeguarding mechanisms for practical robust convergence of the method and can also serve as base for the selection of the penalty parameter for other bilevel optimization algorithms. We also compare the Levenberg–Marquardt method introduced in this paper to other existing algorithms of similar nature.