A new numerical method is presented for bilevel programs with a nonconvex follower’s problem. The basic idea is to piecewise construct convex relaxations of the follower’s problems, replace the relaxed follower’s problems equivalently by their Karush–Kuhn–Tucker conditions and solve the resulting mathematical programs with equilibrium constraints. The convex relaxations and needed parameters are constructed with ideas of the piecewise convexity method of global optimization. Under mild conditions, we show that every accumulation point of the optimal solutions of the sequence approximate problems is an optimal solution of the original problem. The convergence theorems of this method are presented and proved. Numerical experiments show that this method is capable of solving this class of bilevel programs.

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具有非凸跟随者问题的双层规划的凸化方法

针对具有非凸跟随者问题的双层规划，提出了一种新的数值方法。基本思想是分段构造跟随者问题的凸松弛，用他们的卡鲁什-库恩-塔克条件等价地替换松弛跟随者的问题，并用平衡约束求解得到的数学程序。凸松弛和所需参数是根据全局优化的分段凸性方法的思想构建的。在温和的条件下，我们证明了序列近似问题的最优解的每个积累点都是原问题的最优解。提出并证明了该方法的收敛定理。数值实验表明，该方法能够求解此类双层程序。