Convexification Method for Bilevel Programs with a Nonconvex Follower’s Problem
Convexification Method for Bilevel Programs with a Nonconvex Follower’s Problem
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DOI：
10.1007/s10957-020-01804-9
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发表时间：
2021-01
影响因子：
1.9
通讯作者：
Gaoxi Li;Xinmin Yang
中科院分区：
文献类型：
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作者：
Gaoxi Li;Xinmin Yang
给出了一种新的求解带非凸跟随者问题的双层规划的数值方法。其基本思想是分段构造从动子问题的凸松弛，用松弛从动子问题的Karush-Kuhn-Tucker条件等价地代替松弛从动子问题，并用平衡约束求解所得到的数学规划。利用全局优化的分段凸性方法的思想，构造了凸松弛函数和所需的参数。在较弱的条件下，证明了序列逼近问题最优解的每个聚点都是原问题的最优解。给出并证明了该方法的收敛定理。数值实验表明，该方法能够求解这类双层规划问题。
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.