Approaches to four types of bilevel programming problems with nonconvex nonsmooth lower level programs and their applications to newsvendor problems
Approaches to four types of bilevel programming problems with nonconvex nonsmooth lower level programs and their applications to newsvendor problems
复制标题
DOI：
10.1007/s00186-017-0592-2
复制
发表时间：
2017-05
影响因子：
1.2
通讯作者：
Xide Zhu;P. Guo
中科院分区：
文献类型：
--
作者：
Xide Zhu;P. Guo
本文主要研究双层规划问题的求解，其中下层规划为极大极小优化问题，上层规划的目标函数为极大极大或极大极小。由于这些双层规划问题包含了非凸、非光滑的下层规划问题，因此是一个具有挑战性的未完成工作。在一定的假设条件下，我们将这些问题转化为一般的单层优化问题或极小极大优化问题。为了处理这些等价的极小极大优化问题，我们提出了一类正则化方法，该方法通过使用一族极大熵函数来逼近极大值函数。此外，我们还研究了所提出的正则化方法的极限情况，并证明了由近似方法获得的全局最优解的任何极限点与原问题的极限点相同。最后，我们将所提出的方法应用到报童问题，并使用数值例子来证明其有效性。
This paper concentrates on solving bilevel programming problems where the lower level programs are max–min optimization problems and the upper level programs have max–max or max–min objective functions. Because these bilevel programming problems include nonconvex and nonsmooth lower level program problems, it is a challenging undone work. Giving some assumptions, we translate these problems into general single level optimization problems or min–max optimization problems. To deal with these equivalent min–max optimization problems, we propose a class of regularization methods which approximate the maximum function by using a family of maximum entropy functions. In addition, we examine the limit situations of the proposed regularization methods and show that any limit points of the global optimal solutions obtained by the approximation methods are the same as the ones of the original problems. Finally, we apply the proposed methods to newsvendor problems and use a numerical example to show their effectiveness.