  In this paper, we consider a class of nonsmooth sum-of-ratios fractional
optimization problems with block structure. This model class is ubiquitous and
encompasses several important nonsmooth optimization problems in the
literature. We first propose an inertial proximal block coordinate method for
solving this class of problems by exploiting the underlying structure. The
global convergence of our method is guaranteed under the Kurdyka--Lojasiewicz
(KL) property and some mild assumptions. We then identify the explicit
exponents of the KL property for three important structured fractional
optimization problems. In particular, for the sparse generalized eigenvalue
problem with either cardinality regularization or sparsity constraint, we show
that the KL exponents are 1/2, and so, the proposed method exhibits linear
convergence rate. Finally, we illustrate our theoretical results with both
analytic and simulated numerical examples.
