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Abstract <p>It is proved that any strongly convex Lipschitz differentiable function satisfies the Polyak–Lojasiewicz condition on a proximally smooth <i>C</i><sup>1</sup>-smooth manifold under a certain relationship between the proximal smoothness constant of the manifold and strong convexity constant of the function. This condition guarantees a linear convergence rate of the gradient projection method for minimizing the function on the manifold. An algorithm for finding a metric projection of a point located sufficiently close to a manifold onto this manifold is proposed.</p>

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The Polyak–Lojasiewicz Condition for a Strongly Convex Function on a Smooth Manifold and Its Application

  • 期刊论文

  • 发表日期：

    2026年4月1日

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  • M. V. Balashov

摘要

Abstract

It is proved that any strongly convex Lipschitz differentiable function satisfies the Polyak–Lojasiewicz condition on a proximally smooth C1-smooth manifold under a certain relationship between the proximal smoothness constant of the manifold and strong convexity constant of the function. This condition guarantees a linear convergence rate of the gradient projection method for minimizing the function on the manifold. An algorithm for finding a metric projection of a point located sufficiently close to a manifold onto this manifold is proposed.