  This paper is concerned with a class of nonmonotone descent methods for
minimizing a proper lower semicontinuous KL function $\Phi$, which generates a
sequence satisfying a nonmonotone decrease condition and a relative error
tolerance. Under suitable assumptions, we prove that the whole sequence
converges to a limiting critical point of $\Phi$ and, when $\Phi$ is a KL
function of exponent $\theta\in[0,1)$, the convergence admits a linear rate if
$\theta\in[0,1/2]$ and a sublinear rate associated to $\theta$ if
$\theta\in(1/2,1)$. The required assumptions are shown to be sufficient and
necessary if $\Phi$ is also weakly convex on a neighborhood of stationary point
set. Our results resolve the convergence problem on the iterate sequence
generated by a class of nonmonotone line search algorithms for nonconvex and
nonsmooth problems, and also extend the convergence results of monotone descent
methods for KL optimization problems. As the applications, we achieve the
convergence of the iterate sequence for the nonmonotone line search proximal
gradient method with extrapolation and the nonmonotone line search proximal
alternating minimization method with extrapolation. Numerical experiments are
conducted for zero-norm and column $\ell_{2,0}$-norm regularized problems to
validate their efficiency.
