Open AccessTowards Supremum-Sum Subdifferential Calculus Free of Qualification Conditions
Author(s) -
Rafaël Correa,
Abderrahim Hantoute,
Marco A. López
Publication year - 2016
Publication title -
siam journal on optimization
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.066
H-Index - 136
eISSN - 1095-7189
pISSN - 1052-6234
DOI - 10.1137/15m1045375
Subject(s) - subderivative , mathematics , infimum and supremum , convex function , regular polygon , function (biology) , convex analysis , convex conjugate , domain (mathematical analysis) , pure mathematics , effective domain , proper convex function , calculus (dental) , mathematical optimization , mathematical analysis , convex optimization , geometry , medicine , dentistry , evolutionary biology , biology
We give a formula for the subdifferential of the sum of two convex functions where one of them is the supremum of an arbitrary family of convex functions. This is carried out under a weak assumption expressing a natural relationship between the lower semicontinuous envelopes of the data functions in the domain of the sum function. We also provide a new rule for the subdifferential of the sum of two convex functions, which uses a strategy of augmenting the involved functions. The main feature of our analysis is that no continuity-type condition is required. Our approach allows us to unify, recover, and extend different results in the recent literature.Research of the first and the second authors is supported by CONICYT grants, Fondecyt 1150909 and 1151003, Basal PFB-03, and Basal FB003. Research of the second and third authors is supported by MINECO of Spain and FEDER of EU, grant MTM2014-59179-C2-1-P. Research of the third author is also supported by the Australian Research Council: Project DP160100854
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