Variational Theory and Algorithms for a Class of Asymptotically Approachable Nonconvex Problems

• 成果类型: Article; Early Access
• 署名作者: Li, Hanyang; Cui, Ying
• 署名单位: University of California System; University of California Berkeley
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2023.0202
• 发表日期: 2025
• 关键词: Optimization; nonsmooth; calculus

摘要

We investigate a class of composite nonconvex functions, where the outer function is the sum of univariate extended-real-valued convex functions and the inner function is the limit of difference-of-convex functions. A notable feature of this class is that the inner function may fail to be locally Lipschitz continuous. It covers a range of important, yet challenging, applications, including inverse optimal value optimization and problems under value-at-risk constraints. We propose an asymptotic decomposition of the composite function that guarantees epi-convergence to the original function, leading to necessary optimality conditions for the corresponding minimization problem. The proposed decomposition also enables us to design a numerical algorithm such that any accumulation point of the generated sequence, if it exists, satisfies the newly introduced optimality conditions. These results expand on the study of so-called amenable functions introduced by Poliquin and Rockafellar in 1992, which are compositions of convex functions with smooth maps, and the prox-linear methods for their minimization. To demonstrate that our algorithmic framework is practically implementable, we further present verifiable termination criteria and preliminary numerical results.