Subdifferential of the supremum function: moving back and forth between continuous and non-continuous settings

• 成果类型: Article
• 署名作者: Correa, R.; Hantoute, A.; Lopez, M. A.
• 署名单位: Universidad de O'Higgins; Universidad de Chile; Universidad de Chile; Universitat d'Alacant; Federation University Australia
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-020-01592-0
• 发表日期: 2021
• 页码: 217-247
• 关键词: calculus rules; upper envelope; convex; formulas; set

摘要

In this paper we establish general formulas for the subdifferential of the pointwise supremum of convex functions, which cover and unify both the compact continuous and the non-compact non-continuous settings. From the non-continuous to the continuous setting, we proceed by a compactification-based approach which leads us to problems having compact index sets and upper semi-continuously indexed mappings, giving rise to new characterizations of the subdifferential of the supremum by means of upper semicontinuous regularized functions and an enlarged compact index set. In the opposite sense, we rewrite the subdifferential of these new regularized functions by using the original data, also leading us to new results on the subdifferential of the supremum. We give two applications in the last section, the first one concerning the nonconvex Fenchel duality, and the second one establishing Fritz-John and KKT conditions in convex semi-infinite programming.