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

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-à 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

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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• http://mi.mathnet.ru/eng/msb9214
• https://doi.org/10.4213/sm9214
• http://mi.mathnet.ru/eng/msb/v211/i4/p3

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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
2. Balashov M.V., “The Gradient Projection Algorithm For Smooth Sets and Functions in Nonconvex Case”, Set-Valued Var. Anal.
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
4. M. V. Balashov, “Growth Conditions on a Function and the Error Bound Condition”, Math. Notes, 109:4 (2021), 638–643
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