  Minimization of a smooth function on a sphere or, more generally, on a smooth
manifold, is the simplest non-convex optimization problem. It has a lot of
applications. Our goal is to propose a version of the gradient projection
algorithm for its solution and to obtain results that guarantee convergence of
the algorithm under some minimal natural assumptions. We use the
Lezanski-Polyak-Lojasiewicz condition on a manifold to prove the global linear
convergence of the algorithm. Another method well fitted for the problem is the
conditional gradient (Frank-Wolfe) algorithm. We examine some conditions which
guarantee global convergence of full-step version of the method with linear
rate.
