
Journal of Convex Analysis 30 (2023), No. 4, 11731201 Copyright Heldermann Verlag 2023 Combined Approach with SecondOrder 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 secondorder 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 firstorder and the value function approaches to derive new necessary optimality conditions. In our approach, we add a secondorder optimality condition to the combined program as a new constraint. We show that when all known approaches fail, adding the secondorder 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 firstorder and the secondorder information. Keywords: Partial calmness, bilevel program, optimality condition, secondorder optimality condition. MSC: 90C26, 90C30, 90C31, 90C33, 90C46, 49J52, 91A65. [ Fulltextpdf (263 KB)] for subscribers only. 