The Polyak-Lojasiewicz condition for a Lipschitz differentiable function on a smooth manifold

The Polyak-Lojasiewicz condition in unconstrained minimization ensures convergence with the rate of geometric progression of the gradient descent method, random coordinate descent, and a number of other algorithms for Lipschitz-differentiable and, in general, nonconvex functions. This condition is also closely related to some other properties of the function being minimized. We shall discuss a similar property for a Lipschitz differentiable function on a smooth compact manifold. The relationship with other conditions and the rate of convergence of the gradient projection method will be considered.