This paper considers regularized block multiconvex optimization, where the feasible set and objective
function are generally nonconvex but convex in each block of variables. It also accepts nonconvex
blocks and requires these blocks to be updated by proximal minimization. We review some interesting
applications and propose a generalized block coordinate descent method. Under certain
conditions, we show that any limit point satisfies the Nash equilibrium conditions. Furthermore, we
establish global convergence and estimate the asymptotic convergence rate of the method by assuming
a property based on the Kurdyka-Lojasiewicz inequality. The proposed algorithms are tested on
nonnegative matrix and tensor factorization, as well as matrix and tensor recovery from incomplete
observations. The tests include synthetic data and hyperspectral data, as well as image sets from
the CBCL and ORL databases. Compared to the existing state-of-the-art algorithms, the proposed
algorithms demonstrate superior performance in both speed and solution quality. The MATLAB
code of nonnegative matrix/tensor decomposition and completion, along with a few demos, are
accessible from the authors' homepages.