Authors: Stošić, Marko 
Xavier, João
Dodig, Marija 
Title: Projection on the intersection of convex sets
Journal: Linear Algebra and Its Applications
Volume: 509
First page: 191
Last page: 205
Issue Date: 15-Nov-2016
Rank: M21
ISSN: 0024-3795
DOI: 10.1016/j.laa.2016.07.023
Abstract: 
In this paper, we give a solution of the problem of projecting a point onto the intersection of several closed convex sets, when a projection on each individual convex set is known. The existing solution methods for this problem are sequential in nature. Here, we propose a highly parallelizable method. The key idea in our approach is the reformulation of the original problem as a system of semi-smooth equations. The benefits of the proposed reformulation are twofold: (a) a fast semi-smooth Newton iterative technique based on Clarke's generalized gradients becomes applicable and (b) the mechanics of the iterative technique is such that an almost decentralized solution method emerges. We proved that the corresponding semi-smooth Newton algorithm converges near the optimal point (quadratically). These features make the overall method attractive for distributed computing platforms, e.g. sensor networks.
Keywords: Generalized Jacobian | Projections | Semi-smooth Newton algorithm
Publisher: Elsevier
Project: Geometry and Topology of Manifolds, Classical Mechanics and Integrable Dynamical Systems 
Geometry, Education and Visualization With Applications 
Fundação para a Ciência e a Tecnologia, projects ISFL-1-1431, PTDC/EMS-CRO/2042/2012 and UID/EEA/5009/2013

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