The Lezanski – Polyak – Lojasiewicz inequality and the convergence of the gradient projection algorithm

We consider the Lezanski – Polyak – Lojasiewicz inequality for a real-analytic function on a real-analytic compact manifold without boundary in finite-dimensional Euclidean space. This inequality emerged in 1963 independently in works of three authors: Lezanski and Lojasiewicz from Poland and Polyak from the USSR. The inequality is appeared to be a very useful tool in the convergence analysis of the gradient methods, firstly in unconstrained optimization and during the past few decades in problems of constrained optimization. Basically, it is applied for a smooth in a certain sense function on a smooth in a certain sense manifold. We propose the derivation of the inequality from the error bound condition of the power type on a compact real-analytic manifold. As an application, we prove the convergence of the gradient projection algorithm of a real analytic function on a real analytic compact manifold without boundary. Unlike known results, our proof gives explicit dependence of the error via parameters of the problem: the power in the error bound condition and the constant of proximal smoothness first of all. Here we significantly use a technical fact that a smooth compact manifold without boundary is a proximally smooth set.