joint work with Martin Schmidt
Bilevel problems are highly challenging optimization problems that appear in many applications. Due to the hierarchical structure, they are inherently non-convex. In this talk we present a primal heuristic for (MI)QP-QP bilevel problems based on a penalty alternating direction method. We derive a convergence theory stating that the method converges to a stationary point of an equivalent single-level reformulation of the bilevel problem. Further, we provide an extensive numerical study and illustrate the good performance of our method both in terms of running times and solution quality.
We consider here the mathematical program with complementarity constraints (MPCC). In a recent work by Dussault et al. [How to compute an M-stationary point of the MPCC], the authors present an algorithmic scheme based on a general family of regularization for the MPCC with guaranteed convergence to an M-stationary point. This algorithmic scheme is a regularization-penalization-active set approach, and thus, regroups several techniques from nonlinear optimization. Our aim in this talk is to discuss this algorithmic scheme and several aspects of its implementation in Julia.
This talk considers a class of two-stage stochastic linear variational inequality problems. We first give conditions for the existence of solutions to both the original problem and its perturbed problems. Next we derive quantitative stability assertions of this two-stage stochastic problem under total variation metrics via the corresponding residual function. After that, we study the discrete approximation problem. The convergence and the exponential rate of convergence of optimal solution sets are obtained under moderate assumptions respectively.