Xiaoxiao Ma

Topic

On Bilevel Programs and Minimax Problems

Department of Mathematics and Statistics

Date & location

• Monday, August 12, 2024
• 9:00 A.M.
• Virtual Defence

Examining Committee

Supervisory Committee

• Dr. Jane Ye, Department of Mathematics and Statistics, University of Victoria (Supervisor)
• Dr. Julie Zhou, Department of Mathematics and Statistics, UVic (Member)
• Dr. Yang Shi, Department of Mechanical Engineering, UVic (Outside Member)

External Examiner

• Dr. Patrick Mehlitz, Department of Mathematics and Computer Science, University of Marburg

Chair of Oral Examination

• Dr. Tao Wang, Department of Economics, UVic

Abstract

Second-order optimality conditions usually offer more precise insights into local optimality compared to their first-order counterparts. Concurrently, there has been a growing prevalence of bilevel programs and minimax problems in recent years. In our research, we intricately explore second-order optimality conditions within the realm of bilevel programs and minimax problems.

First, we provide a comprehensive exploration of second-order combined approaches for bilevel problems. Building on the well-known first-order combined approach, the research introduces novel techniques that incorporate lower-level second-order information to overcome the difficulty of the constraint qualification for bilevel problems. By characterizing lower-level optimal solutions using both first and second-order necessary optimality conditions, together with the value function constraint, we give some new single-level reformulations for bilevel problems for which the important partial calmness condition can be more likely to hold.

We then focus on the introduction and analysis of calm local minimax points, which is an appropriate local notion for nonconvex-nonconcave nonsmooth minimax problems. We study the properties of calm local minimax points, establishing their strong connections with existing optimality concepts. We provide a comprehensive exploration of first-order and second-order sufficient and necessary optimality conditions for calm local minimax points.