Journal of Convex Analysis 30 (2023), No. 4, 1173--1201
Copyright Heldermann Verlag 2023
Combined Approach with Second-Order Optimality Conditions for Bilevel Programming Problems
Xiaoxiao Ma
Dept. of Mathematics and Statistics, University of Victoria, Canada
xiaoxiaoma@uvic.ca
Wei Yao
Dept. of Mathematics, Southern University of Science and Technology, and: National Center for Applied Mathematics, Shenzhen, Guangdong, China
yaow@sustech.edu.cn
Jane J. Ye
Dept. of Mathematics and Statistics, University of Victoria, Canada
janeye@uvic.ca
Jin Zhang
Dept. of Mathematics and SUSTech International Center for Mathematics, Southern University of Science and Technology, and: National Center for Applied Mathematics, Shenzhen, Guangdong, China
zhangj9@sustech.edu.cn
We propose a combined approach with second-order optimality conditions of the lower level problem to study constraint qualifications and optimality conditions for bilevel programming problems. The new method is inspired by the combined approach developed by Ye and Zhu in 2010, where the authors combined the classical first-order and the value function approaches to derive new necessary optimality conditions. In our approach, we add a second-order optimality condition to the combined program as a new constraint. We show that when all known approaches fail, adding the second-order optimality condition as a constraint makes the corresponding partial calmness condition and the resulting necessary optimality condition easier to hold. We also give some discussions on advantages and disadvantages of the combined approaches with the first-order and the second-order information.
Keywords: Partial calmness, bilevel program, optimality condition, second-order optimality condition.
MSC: 90C26, 90C30, 90C31, 90C33, 90C46, 49J52, 91A65.
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