Determination of convex functions via subgradients of minimal norm

• 成果类型: Article
• 署名作者: Perez-Aros, Pedro; Salas, David; Vilches, Emilio
• 署名单位: Universidad de O'Higgins; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universidad de Chile
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-020-01550-w
• 发表日期: 2021
• 页码: 561-583
• 关键词: integration; subdifferentials

摘要

We show, in Hilbert space setting, that any two convex proper lower semicontinuous functions bounded from below, for which the norm of their minimal subgradients coincide, they coincide up to a constant. Moreover, under classic boundary conditions, we provide the same results when the functions are continuous and defined over an open convex domain. These results show that for convex functions bounded from below, the slopes provide sufficient first-order information to determine the function up to a constant, giving a positive answer to the conjecture posed in Boulmezaoud et al. (SIAM J Optim 28(3):2049-2066, 2018) .