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Projection methods in conic optimization
There exist efficient algorithms to project a point onto the intersection of
a convex cone and an affine subspace. Those conic projections are in turn the
work-horse of a range of algorithms in conic optimization, having a variety of
applications in science, finance and engineering. This chapter reviews some of
these algorithms, emphasizing the so-called regularization algorithms for
linear conic optimization, and applications in polynomial optimization. This is
a presentation of the material of several recent research articles; we aim here
at clarifying the ideas, presenting them in a general framework, and pointing
out important techniques
Nonconvex notions of regularity and convergence of fundamental algorithms for feasibility problems
We consider projection algorithms for solving (nonconvex) feasibility
problems in Euclidean spaces. Of special interest are the Method of Alternating
Projections (MAP) and the Douglas-Rachford or Averaged Alternating Reflection
Algorithm (AAR). In the case of convex feasibility, firm nonexpansiveness of
projection mappings is a global property that yields global convergence of MAP
and for consistent problems AAR. Based on (\epsilon, \delta)-regularity of sets
developed by Bauschke, Luke, Phan and Wang in 2012, a relaxed local version of
firm nonexpansiveness with respect to the intersection is introduced for
consistent feasibility problems. Together with a coercivity condition that
relates to the regularity of the intersection, this yields local linear
convergence of MAP for a wide class of nonconvex problems,Comment: 22 pages, no figures, 30 reference
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