Elsevier

Linear Algebra and its Applications

Volume 509, 15 November 2016, Pages 191-205
Linear Algebra and its Applications

Projection on the intersection of convex sets

https://doi.org/10.1016/j.laa.2016.07.023Get rights and content
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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.

MSC

90C53
15A09

Keywords

Projections
Semi-smooth Newton algorithm
Generalized Jacobian

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