We consider a class of optimistic bilevel problems. Specifically, we address bilevel problems in which at the lower level the objective function is fully convex and the feasible set does not depend on the upper level variables. We show that this nontrivial class of mathematical programs is sufficiently broad to encompass significant real-world applications and proves to be numerically tractable. From this respect, we establish that the stationary points for a relaxation of the original problem can be obtained addressing a suitable generalized Nash equilibrium problem. The latter game is proven to be convex and with a nonempty solution set. Leveraging this correspondence, we provide a provably convergent, easily implementable scheme to calculate stationary points of the relaxed bilevel program. As witnessed by some numerical experiments on an application in economics, this algorithm turns out to be numerically viable also for big dimensional problems.

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数值可处理的乐观双层问题

我们考虑一类乐观的双层问题。具体而言，我们解决了以下两级问题：在较低级别上，目标函数是完全凸的，而可行集不依赖于较高级别的变量。我们证明了这一非平凡的数学程序类别足够广泛，足以涵盖重要的现实应用，并且在数值上证明是易处理的。从这一方面出发，我们确定可以获得松弛的原始点，可以解决合适的广义纳什均衡问题。后者被证明是凸的，并且具有非空解集。利用这种对应关系，我们提供了一种可证明是收敛的，易于实现的方案来计算松弛双层程序的固定点。