  Light emitting diode (LED) based visible light positioning (VLP) networks can
provide accurate location information in indoor environments. In this
manuscript, we propose to employ cooperative localization for visible light
networks by designing a VLP system configuration that involves multiple LED
transmitters with known locations (e.g., on the ceiling) and visible light
communication (VLC) units equipped with both LEDs and photodetectors (PDs) for
the purpose of cooperation. First, we derive the Cram\'er-Rao lower bound
(CRLB) and the maximum likelihood estimator (MLE) for the localization of VLC
units in the proposed cooperative scenario. To tackle the nonconvex structure
of the MLE, we adopt a set-theoretic approach by formulating the problem of
cooperative localization as a quasiconvex feasibility problem, where the aim is
to find a point inside the intersection of convex constraint sets constructed
as the sublevel sets of quasiconvex functions resulting from the Lambertian
formula. Next, we devise two feasibility-seeking algorithms based on iterative
gradient projections to solve the feasibility problem. Both algorithms are
amenable to distributed implementation, thereby avoiding high-complexity
centralized approaches. Capitalizing on the concept of quasi-Fej\'er convergent
sequences, we carry out a formal convergence analysis to prove that the
proposed algorithms converge to a solution of the feasibility problem in the
consistent case. Numerical examples illustrate the improvements in localization
performance achieved via cooperation among VLC units and evidence the
convergence of the proposed algorithms to true VLC unit locations in both the
consistent and inconsistent cases.
