Restricted Normal Cones and the Method of Alternating Projections: Theory

By Heinz H. Bauschke, D. R. Luke, Hung M. Phan, Xianfu Wang. 2012

78 citations recorded by the source corpus.

In this paper, we introduce and develop the theory of restricted normal cones which generalize the classical Mordukhovich normal cone. We thoroughly study these objects from the viewpoint of constraint qualifications and regularity. Numerous examples are provided to illustrate the theory. This work provides the theoretical underpinning for a subsequent article in which these tools are applied to obtain a convergence analysis of the method of alternating projections for nonconvex sets.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/e9c60ea62f0d7934f0a51f999e56af7bb7e66843

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In Computational Theory and Mathematics

In Geometry and Topology

In Numerical Analysis

Citations in the corpus

Restricted Normal Cones and the Method of Alternating Projections: Applications (2012, methodology): “We use the technology of restricted normal cones developed in [6] to develop the least restrictive sufficient conditions to date for linear convergence of the sequences generated by the method of alternating projections applied to two sets A and B.”
On the local convergence of the Douglas–Rachford algorithm (2014, methodology): “Without convexity convergence of the DR-projection algorithm and of its companion, the method of alternating projections (MAP) are usually based on transversality assumptions, see [4] for MAP, and [11] for DR.”
Linear convergence of the Douglas–Rachford method for two closed sets (2014, unknown): “d amenability or prox-regularity. This concept is further generalized in [13, Deﬁnition8.1]and[22,Deﬁnition2.9]. Importantly,allconvexsetsaresuperregular. More discussion and examples can be found in [13, 22, 27]. Deﬁnition 2.7 (γ-quasi ﬁrm expansiveness) A mapping Φ : X ⇒ X is said to be (Ω,γ)- qua”