  This paper proposes and develops new linesearch methods with inexact gradient
information for finding stationary points of nonconvex continuously
differentiable functions on finite-dimensional spaces. Some abstract
convergence results for a broad class of linesearch methods are stablished. A
general scheme for inexact reduced gradient (IRG) methods is proposed, where
the errors in the gradient approximation automatically adapt with the
magnitudes of the exact gradients. The sequences of iterations are shown to
obtain stationary accumulation points when different stepsize selections are
employed. Convergence results with constructive convergence rates for the
developed IRG methods are established under the Kurdyka- Lojasiewicz property.
The obtained results for the IRG methods are confirmed by encouraging numerical
experiments, which demonstrate advantages of automatically controlled errors in
IRG methods over other frequently used error selections.
