Abstract. Some properties of regularized and penalized nonlinear programming formulations of mathematical programs with equilibrium constraints (MPECs) are described. The focus is on the properties of these formulations near a local solution of the MPEC at which strong stationarity and a second-order sufficient condition are satisfied. In the regularized formulations, the complementarity condition is replaced by a constraint involving a positive parameter that can be decreased to zero. In the penalized formulation, the complementarity constraint appears as a penalty term in the objective. Existence and uniqueness of solutions for these formulations are investigated, and estimates are obtained for the distance of these solutions to the MPEC solution under various assumptions.

Abstract. The successful application of Support Vector Machines (SVMs), kernel methods and other statistical machine learning methods requires selection of model parameters based on estimates of the generalization error. This paper presents a novel approach to systematic model selection through bilevel optimization. We show how modelling tasks for widely used machine learning methods can be formulated as bilevel optimization problems and describe how the approach can address a broad range of tasks—among which are parameter, feature and kernel selection In addition, we also discuss the challenges in implementing these approaches and enumerate opportunities for future work in this emerging research area. 1.