Abstract
In this paper, we develop new subgradient methods for solving nonsmooth convex optimization problems. These methods guarantee the best possible rate of convergence for the whole sequence of test points. Our methods are applicable as efficient real-time stabilization tools for potential systems with infinite horizon. Preliminary numerical experiments confirm a high efficiency of the new schemes.
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Notes
For the sake of notation, we assume that this subgradient is uniquely defined by the argument. At the points of nondifferentiability, this, in general, is not true. However, we assume that at such points the first-order oracle always returns the same answer. The same convention is used for all convex functions in this paper (e.g. prox-functions; see below).
In paper [5], Beck and Teboulle justified a primal subgradient method, which works with Bregman distances: \(x_{t+1} = \min \limits _{x \in Q} \{ a_t \langle \nabla f(x_t), x \rangle + D(x_t,x) \}\), where \(D(x,y) = d(y) - d(x) - \langle \nabla d(x), y - x \rangle \). The rate of convergence of this method can be derived from the same inequality (10). In our terminology, this is a pure primal scheme since it does not maintain a linear model of the objective function.
Recall that, for Euclidean framework with \(Q \equiv \mathbb {E}\), PGM coincides with MDM (9).
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Acknowledgments
The authors would like to thank two anonymous referees for careful reading and useful comments. Research results presented in the paper have been supported by a Grant “Action de recherche concertè ARC 04/09-315” from the “Direction de la recherche scientifique - Communautè française de Belgique” and by PremoLab, MIPT (RF gov. Grant, ag.11.G34.31.0073), with RFBR research Projects 13-01-12007 ofi_m, 14-01-00722-a.
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Communicated by Amir Beck.
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Nesterov, Y., Shikhman, V. Quasi-monotone Subgradient Methods for Nonsmooth Convex Minimization. J Optim Theory Appl 165, 917–940 (2015). https://doi.org/10.1007/s10957-014-0677-5
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DOI: https://doi.org/10.1007/s10957-014-0677-5