2025/02/24 by Xiang Liu, Mengwei Xu, Liwei Zhang
Computer Science · Engineering · Mathematics · #Applied mathematics #Bilevel optimization #Business #Fixed Point Theorems Analysis #Mathematical economics #Mathematical optimization #Mathematics #Optimization and Mathematical Programming #Optimization and Variational Analysis #Optimization problem #Order (exchange)
paper · doi:10.1080/02331934.2025.2468412
crossref issued 2025/02/24 · crossref published 2025/02/24 · crossref published-online 2025/02/24 · openalex publication_date 2025/02/24 · crossref created 2025/02/25 · crossref published-print 2025/09/10 · openalex created_date 2025/10/10 · crossref deposited 2026/02/09 · crossref indexed 2026/08/03 · openalex updated_date 2026/08/04
A popular way for developing optimality conditions of bilevel programs is to express them as (single-level) constrained optimization problems involving optimal value functions of lower programs, in which the computation of first/second-order directional derivatives for the value functions or solution mappings of the lower level problems is a challenging task. This type of directional derivatives based optimality conditions are not suitable for analysing theoretical properties of numerical algorithms for bilevel programs, especially when lower programs are non-convex optimization problems. This paper focuses on bilevel programs whose lower level optimization problems are non-convex. We introduce the concept of bi-local solutions and prove that, under the Jacobian uniqueness conditions on the lower level problem, the bi-local solutions are just the local solutions of the optimization problem constrained by Karush–Kuhn–Tucker (KKT) conditions of the lower level problem. Importantly, under suitable conditions, we prove that the KKT constrained optimization problem satisfies the Mangasarian–Fromovitz constraint qualification (MFCQ). Based on the equivalence of bi-local solutions and local solutions of the KKT constrained optimization problem, we establish the second-order necessary and sufficient optimality conditions for the bi-local optimal solutions. Different from second-order optimality conditions in terms of second-order directional derivatives of value functions, the second-order optimality conditions in this paper rely only on the second-order derivatives of the problem functions of the bilevel program. Moreover, the second-order sufficient optimality conditions can be applied for analysing convergence properties of numerical algorithms for the bilevel program.