Subdifferential of the supremum function: moving back and forth between continuous and non-continuous settings
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Subdifferential of the supremum function: moving back and forth between continuous and non-continuous settings R. Correa1,2 · A. Hantoute3,4
· M. A. López4,5
Received: 23 January 2020 / Accepted: 3 November 2020 © Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2020
Abstract 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. ˇ Keywords Supremum of convex functions · Subdifferentials · Stone–Cech compactification · Convex semi-infinite programming · Fritz-John and KKT optimality conditions Mathematics Subject Classification 46N10 · 52A41 · 90C25
1 Introduction In this paper we deal with the characterization of the subdifferential of the pointwise supremum f := supt∈T f t of a family of convex functions f t : X → R ∪ {±∞},
Dedicated by his co-authors to Prof. Macro A. López on his 70th birthday Research supported by CONICYT (Fondecyt 1190012 and 1190110), Proyecto/Grant PIA AFB-170001, MICIU of Spain and Universidad de Alicante (Grant Beatriz Galindo BEAGAL 18/00205), and Research Project PGC2018-097960-B-C21 from MICINN, Spain. The research of the third author is also supported by the Australian ARC - Discovery Projects DP 180100602. Extended author information available on the last page of the article
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t ∈ T , with T being an arbitrary nonempty set, defined on a separated locally convex space X . We obtain new characterizations which allow us to unify both the compact continuous and the non-compact non-continuous setting ([8,9,27,30], etc.). The first setting relies on the following standard conditions in the literature of convex analysis and non-differentiable semi-infinite programming: T is compact and the mappings f (·) (z), z ∈ X , are upper semi-continuous. In the other framework, called the non-compact non-continuous setting, we do not assume the above conditions. In other words (see, e.g., [14,15,18,21,29–31], etc.): T is an arbitrary set, possibly infinite and without any prescribed topology, and no requirement is imposed on the mappings f (·) (z). Going from the non-continuous to the continuous setting, we follow an approach ˇ bas
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