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Mat. Sb., 2020, Volume 211, Number 4, Pages 3–26 (Mi msb9214)  

This article is cited in 4 scientific papers (total in 4 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.

Funding Agency Grant Number
Russian Foundation for Basic Research 18-01-00209-а
This research was carried out with the financial support of the Russian Foundation for Basic Research (grant no. 18-01-00209-a).


DOI: https://doi.org/10.4213/sm9214

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

Bibliographic databases:

UDC: 519.853.651+517.982+519.853.4
MSC: Primary 90C26, 49J53; Secondary 46N10, 65K10
Received: 09.01.2019 and 13.08.2019

Citation: M. V. Balashov, “The gradient projection algorithm for a proximally smooth set and a function with Lipschitz continuous gradient”, Mat. Sb., 211:4 (2020), 3–26; Sb. Math., 211:4 (2020), 481–504

Citation in format AMSBIB
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\by M.~V.~Balashov
\paper The gradient projection algorithm for a~proximally smooth set and a~function with Lipschitz continuous gradient
\jour Mat. Sb.
\yr 2020
\vol 211
\issue 4
\pages 3--26
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\jour Sb. Math.
\yr 2020
\vol 211
\issue 4
\pages 481--504
\crossref{https://doi.org/10.1070/SM9214}
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Balashov V M., Tremba A.A., “Error Bound Conditions and Convergence of Optimization Methods on Smooth and Proximally Smooth Manifolds”, Optimization  crossref  isi
    2. Balashov M.V., “The Gradient Projection Algorithm For Smooth Sets and Functions in Nonconvex Case”, Set-Valued Var. Anal.  crossref  isi
    3. M. V. Balashov, “On the Gradient Projection Method for Weakly Convex Functions on a Proximally Smooth Set”, Math. Notes, 108:5 (2020), 643–651  mathnet  crossref  crossref  mathscinet  isi  elib
    4. M. V. Balashov, “Growth Conditions on a Function and the Error Bound Condition”, Math. Notes, 109:4 (2021), 638–643  mathnet  crossref  crossref  isi
  • Математический сборник Sbornik: Mathematics (from 1967)
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