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Transversality and Alternating Projections for Nonconvex Sets

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Abstract

We consider the method of alternating projections for finding a point in the intersection of two closed sets, possibly nonconvex. Assuming only the standard transversality condition (or a weaker version thereof), we prove local linear convergence. When the two sets are semi-algebraic and bounded, but not necessarily transversal, we nonetheless prove subsequence convergence.

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Notes

  1. [34, Remark 10] states that “intrinsic transversality [the property, weaker than transversality, that we actually use to guarantee linear convergence] amalgamates transversality and regularity aspects”. However, that statement concerns “regularity” aspects that involve both sets, like the idea of “linear regularity” in [3].

  2. The proof of [33, Proposition 8] states: “Following entirely the argument in [14, p.6]...”.

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Correspondence to A. S. Lewis.

Additional information

Communicated by Michael Todd.

D. Drusvyatskiy’s research supported in part by AFOSR YIP FA9550-15-1-0237.

A.D. Ioffe’s research supported in part by the US–Israel Binational Science Foundation Grant 2008261.

A.S. Lewis’s research supported in part by National Science Foundation Grant DMS-1208338 and by the US–Israel Binational Science Foundation Grant 2008261.

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Drusvyatskiy, D., Ioffe, A.D. & Lewis, A.S. Transversality and Alternating Projections for Nonconvex Sets. Found Comput Math 15, 1637–1651 (2015). https://doi.org/10.1007/s10208-015-9279-3

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