# Gradient projection and conditional gradient methods for constrained nonconvex minimization

## TL;DR

This paper introduces gradient projection and conditional gradient methods for solving smooth non-convex minimization problems on manifolds, providing convergence guarantees under minimal assumptions and demonstrating their effectiveness.

## Contribution

It develops and analyzes gradient projection and Frank-Wolfe algorithms with convergence guarantees for constrained non-convex optimization on manifolds.

## Key findings

- Gradient projection method converges linearly under Lezanski-Polyak-Lojasiewicz condition.
- Conditional gradient method achieves global convergence with linear rate under certain conditions.
- The methods are applicable to optimization problems on spheres and smooth manifolds.

## Abstract

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.