Error bound conditions and convergence of optimization methods on smooth and proximally smooth manifolds

TL;DR

This paper investigates the convergence properties of a combined gradient projection and Newton method for nonconvex constrained optimization on smooth and proximally smooth manifolds, introducing new error bound conditions.

Contribution

It proposes novel error bound conditions that establish the convergence domain for the Newton method in nonconvex constrained optimization problems.

Findings

New error bound conditions are typical for a wide class of problems.

High convergence rates are achievable by switching to the Newton method.

Convergence to stationary points is guaranteed under the proposed conditions.

Abstract

We analyse the convergence of the gradient projection algorithm, which is finalized with the Newton method, to a stationary point for the problem of nonconvex constrained optimization $\min_{x \in S} f(x)$ with a proximally smooth set $S = \{x \in R^n : g(x) = 0 \}, \; g : R^n \rightarrow R^m$ and a smooth function $f$. We propose new Error bound (EB) conditions for the gradient projection method which lead to the convergence domain of the Newton method. We prove that these EB conditions are typical for a wide class of optimization problems. It is possible to reach high convergence rate of the algorithm by switching to the Newton method.