  We consider in this paper a class of single-ratio fractional minimization
problems, in which the numerator part of the objective is the sum of a
nonsmooth nonconvex function and a smooth nonconvex function while the
denominator part is a nonsmooth convex function. In this work, we first derive
its first-order necessary optimality condition, by using the first-order
operators of the three functions involved. Then we develop first-order
algorithms, namely, the proximity-gradient-subgradient algorithm (PGSA), PGSA
with monotone line search (PGSA_ML) and PGSA with nonmonotone line search
(PGSA_NL). It is shown that any accumulation point of the sequence generated by
them is a critical point of the problem under mild assumptions. Moreover, we
establish global convergence of the sequence generated by PGSA or PGSA_ML and
analyze its convergence rate, by further assuming the local Lipschitz
continuity of the nonsmooth function in the numerator part, the smoothness of
the denominator part and the Kurdyka- Lojasiewicz property of the objective.
The proposed algorithms are applied to the sparse generalized eigenvalue
problem associated with a pair of symmetric positive semidefinite matrices and
the corresponding convergence results are obtained according to their general
convergence theorems. We perform some preliminary numerical experiments to
demonstrate the efficiency of the proposed algorithms
