arXiv subjects

Maxim Balashov

Publications and source records attributed to Maxim Balashov.

2 recordsLinked to original sources

arXiv2019

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

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.

math.OC↗

arXiv2019

Gradient projection and conditional gradient methods for constrained nonconvex minimization

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.

math.OC↗