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JOURNALS // Matematicheskii Sbornik // Archive

Mat. Sb., 2020 Volume 211, Number 4, Pages 3–26 (Mi sm9214)

This article is cited in 6 papers

The gradient projection algorithm for a proximally smooth set and a function with Lipschitz continuous gradient

M. V. Balashov

V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, Russia

Abstract: We consider the minimization problem for a nonconvex function with Lipschitz continuous gradient on a proximally smooth (possibly nonconvex) subset of a finite-dimensional Euclidean space. We introduce the error bound condition with exponent $\alpha\in(0,1]$ for the gradient mapping. Under this condition, it is shown that the standard gradient projection algorithm converges to a solution of the problem linearly or sublinearly, depending on the value of the exponent $\alpha$. This paper is theoretical.
Bibliography: 23 titles.

Keywords: gradient projection algorithm, gradient mapping, error bound condition, proximal smoothness, nonconvex extremal problem.

UDC: 519.853.651+517.982+519.853.4

MSC: Primary 90C26, 49J53; Secondary 46N10, 65K10

Received: 09.01.2019 and 13.08.2019

DOI: 10.4213/sm9214


 English version:
Sbornik: Mathematics, 2020, 211:4, 481–504

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© Steklov Math. Inst. of RAS, 2025