On sums and convex combinations of projectors onto convex sets

2018/02/07 by Bauschke, Heinz H., Bui, Minh N., Wang, Xianfu
#46A32 #47H05 #47H09 #47L07 (Primary) #52A05 #52A41 #90C25 (Secondary) #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.1802.02287

Abstract

The projector onto the Minkowski sum of closed convex sets is generally not equal to the sum of individual projectors. In this work, we provide a complete answer to the question of characterizing the instances where such an equality holds. Our results unify and extend the case of linear subspaces and Zarantonello's results for projectors onto cones. A detailed analysis in the case of convex combinations is also carried out. We establish the partial sum property for projectors onto convex cones, and we also present various examples as well as a detailed analysis in the univariate case.

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