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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2014, Volume 284, Pages 252–270
DOI: https://doi.org/10.1134/S037196851401018X
(Mi tm3531)
 

This article is cited in 10 scientific papers (total in 10 papers)

Greedy expansions in convex optimization

V. N. Temlyakovab

a Mathematics Department, University of South Carolina, Columbia, SC 29208, USA
b Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow, Russia
References:
Abstract: This paper is a follow-up to the author's previous paper on convex optimization. In that paper we began the process of adjusting greedy-type algorithms from nonlinear approximation for finding sparse solutions of convex optimization problems. We modified there the three most popular greedy algorithms in nonlinear approximation in Banach spaces – Weak Chebyshev Greedy Algorithm, Weak Greedy Algorithm with Free Relaxation, and Weak Relaxed Greedy Algorithm – for solving convex optimization problems. We continue to study sparse approximate solutions to convex optimization problems. It is known that in many engineering applications researchers are interested in an approximate solution of an optimization problem as a linear combination of elements from a given system of elements. There is an increasing interest in building such sparse approximate solutions using different greedy-type algorithms. In this paper we concentrate on greedy algorithms that provide expansions, which means that the approximant at the mth iteration is equal to the sum of the approximant from the previous, (m1)th, iteration and one element from the dictionary with an appropriate coefficient. The problem of greedy expansions of elements of a Banach space is well studied in nonlinear approximation theory. At first glance the setting of a problem of expansion of a given element and the setting of the problem of expansion in an optimization problem are very different. However, it turns out that the same technique can be used for solving both problems. We show how the technique developed in nonlinear approximation theory, in particular, the greedy expansions technique, can be adjusted for finding a sparse solution of an optimization problem given by an expansion with respect to a given dictionary.
Funding agency Grant number
National Science Foundation DMS-1160841
The research was supported by the NSF grant DMS-1160841.
Received in March 2013
English version:
Proceedings of the Steklov Institute of Mathematics, 2014, Volume 284, Pages 244–262
DOI: https://doi.org/10.1134/S0081543814010180
Bibliographic databases:
Document Type: Article
UDC: 517.518.8
Language: Russian
Citation: V. N. Temlyakov, “Greedy expansions in convex optimization”, Function spaces and related problems of analysis, Collected papers. Dedicated to Oleg Vladimirovich Besov, corresponding member of the Russian Academy of Sciences, on the occasion of his 80th birthday, Trudy Mat. Inst. Steklova, 284, MAIK Nauka/Interperiodica, Moscow, 2014, 252–270; Proc. Steklov Inst. Math., 284 (2014), 244–262
Citation in format AMSBIB
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\paper Greedy expansions in convex optimization
\inbook Function spaces and related problems of analysis
\bookinfo Collected papers. Dedicated to Oleg Vladimirovich Besov, corresponding member of the Russian Academy of Sciences, on the occasion of his 80th birthday
\serial Trudy Mat. Inst. Steklova
\yr 2014
\vol 284
\pages 252--270
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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  • https://www.mathnet.ru/eng/tm3531
  • https://doi.org/10.1134/S037196851401018X
  • https://www.mathnet.ru/eng/tm/v284/p252
  • This publication is cited in the following 10 articles:
    1. Erwan Fouillen, Claire Boyer, Maxime Sangnier, “Proximal boosting: Aggregating weak learners to minimize non-differentiable losses”, Neurocomputing, 520 (2023), 301  crossref
    2. A. V. Dereventsov, V. N. Temlyakov, “Biorthogonal Greedy Algorithms in convex optimization”, Appl. Comput. Harmon. Anal., 60 (2022), 489–511  mathnet  crossref
    3. Wenhui Zhang, Peixin Ye, Shuo Xing, Xu Xu, “Optimality of the Approximation and Learning by the Rescaled Pure Super Greedy Algorithms”, Axioms, 11:9 (2022), 437  crossref
    4. Zh. Gao, G. Petrova, “Rescaled pure greedy algorithm for convex optimization”, Calcolo, 56:2 (2019), UNSP 15  crossref  mathscinet  isi  scopus
    5. H. Nguyen, G. Petrova, “Greedy strategies for convex optimization”, Calcolo, 54:1 (2017), 207–224  crossref  mathscinet  zmath  isi  scopus
    6. R. A. DeVore, V. N. Temlyakov, “Convex optimization on Banach spaces”, Found. Comput. Math., 16:2 (2016), 369–394  crossref  mathscinet  zmath  isi  scopus
    7. V. N. Temlyakov, “Convergence and rate of convergence of some greedy algorithms in convex optimization”, Proc. Steklov Inst. Math., 293 (2016), 325–337  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    8. V. N. Temlyakov, “Dictionary descent in optimization”, Anal. Math., 42:1 (2016), 69–89  crossref  mathscinet  zmath  isi  elib  scopus
    9. V. N. Temlyakov, “Greedy approximation in convex optimization”, Constr. Approx., 41:2 (2015), 269–296  crossref  mathscinet  zmath  isi  elib  scopus
    10. Vladimir Temlyakov, 2014 48th Asilomar Conference on Signals, Systems and Computers, 2014, 1331  crossref
    Citing articles in Google Scholar: Russian citations, English citations
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    Труды Математического института имени В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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