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The Method of Alternating Relaxed Projections for two nonconvex sets
The Method of Alternating Projections (MAP), a classical algorithm for
solving feasibility prob- lems, has recently been intensely studied for
nonconvex sets. However, intrinsically available are only local convergence
results: convergence occurs if the starting point is not too far away from
solutions to avoid getting trapped in certain regions. Instead of taking full
projection steps, it can be advantageous to underrelax, i.e., to move only part
way towards the constraint set, in order to enlarge the regions of convergence.
In this paper, we thus systematically study the Method of Alternating Relaxed
Projections (MARP) for two (possibly nonconvex) sets. Complementing our recent
work on MAP, we es- tablish local linear convergence results for the MARP.
Several examples illustrate our analysis
Approximate gauge symmetry of composite vector bosons
It can be shown in a solvable field theory model that the couplings of the
composite vector bosons made of a fermion pair approach the gauge couplings in
the limit of strong binding. Although this phenomenon may appear accidental and
special to the vector boson made of a fermion pair, we extend it to the case of
bosons being constituents and find that the same phenomenon occurs in more an
intriguing way. The functional formalism not only facilitates computation but
also provides us with a better insight into the generating mechanism of
approximate gauge symmetry, in particular, how the strong binding and global
current conservation conspire to generate such an approximate symmetry. Remarks
are made on its possible relevance or irrelevance to electroweak and higher
symmetries.Comment: Correction of typos. The published versio
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