Bilevel optimization problems have received a lot of attention in the last years and decades. Besides numerous theoretical developments there also evolved novel solution algorithms for mixed-integer linear bilevel problems and the most recent algorithms use branch-and-cut techniques from mixed-integer programming that are especially tailored for the bilevel context. In this paper, we consider MIQP-QP bilevel problems, i.e., models with a mixed-integer convex-quadratic upper level and a continuous convex-quadratic lower level. This setting allows for a strong-duality-based transformation of the lower level which yields, in general, an equivalent nonconvex single-level reformulation of the original bilevel problem. Under reasonable assumptions, we can derive both a multi- and a single-tree outer-approximation-based cutting-plane algorithm. We show finite termination and correctness of both methods and present extensive numerical results that illustrate the applicability of the approaches. It turns out that the proposed methods are capable of solving bilevel instances with several thousand variables and constraints and significantly outperform classical solution approaches.

在过去的几十年中，双层优化问题受到了很多关注。除了众多的理论发展外，还发展了针对混合整数线性双级问题的新颖的求解算法，最新算法使用了混合整数编程中专门针对双级上下文量身定制的分支剪切技术。在本文中，我们考虑了MIQP-QP双层问题，即具有混合整数的凸-二次上层和连续的凸-二次下层的模型。此设置允许对较低级别进行基于强对偶的转换，从而通常生成原始双级问题的等效非凸单级重构。在合理的假设下，我们可以导出基于多树和单树基于外部逼近的割平面算法。我们展示了这两种方法的有限终止和正确性，并给出了广泛的数值结果，说明了这些方法的适用性。事实证明，所提出的方法能够解决具有数千个变量和约束的双层实例，并且明显优于传统的求解方法。