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Extragradient Method in Optimization: Convergence and Complexity

Nguyen, Trong Phong and Pauwels, Edouard and Richard, Emile and Suter, Bruce W. Extragradient Method in Optimization: Convergence and Complexity. (2018) Journal of Optimization Theory and Applications, 176 (1). 137-162. ISSN 0022-3239

(Document in English)

PDF (Author's version) - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader

Official URL: https://doi.org/10.1007/s10957-017-1200-6


We consider the extragradient method to minimize the sum of two functions, the first one being smooth and the second being convex. Under the Kurdyka–Łojasiewicz assumption, we prove that the sequence produced by the extragradient method converges to a critical point of the problem and has finite length. The analysis is extended to the case when both functions are convex. We provide, in this case, a sublinear convergence rate, as for gradient-based methods. Furthermore, we show that the recent small-prox complexity result can be applied to this method. Considering the extragradient method is an occasion to describe an exact line search scheme for proximal decomposition methods. We provide details for the implementation of this scheme for the one-norm regularized least squares problem and demonstrate numerical results which suggest that combining nonaccelerated methods with exact line search can be a competitive choice.

Item Type:Article
Additional Information:Thanks to Springer editor. The original PDF can be found at Journal of Optimization Theory and Applications (ISSN 0022-3239) website: https://link.springer.com/article/10.1007/s10957-017-1200-6
Audience (journal):International peer-reviewed journal
Uncontrolled Keywords:
Institution:French research institutions > Centre National de la Recherche Scientifique - CNRS (FRANCE)
Université de Toulouse > Institut National Polytechnique de Toulouse - Toulouse INP (FRANCE)
Université de Toulouse > Université Toulouse III - Paul Sabatier - UT3 (FRANCE)
Université de Toulouse > Université Toulouse - Jean Jaurès - UT2J (FRANCE)
Université de Toulouse > Université Toulouse 1 Capitole - UT1 (FRANCE)
Other partners > Adroll Grup (USA)
Other partners > United States Air Force - USAF (USA)
Other partners > Universidad de Chile - UCHILE (CHILE)
Laboratory name:
AFRL: Air Force Research Laboratory, Air Force Office of Scientific Research (US) - Fondecyt : National Fund for Scientific and Technological Development (Chili) - Basal Project CMM Universidad de Chile (Chili) - Mines ParisTech (France)
Deposited On:11 Oct 2019 12:29

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