On linear convergence of a distributed dual gradient algorithm for linearly constrained separable convex problems
In this paper we propose a fully distributed dual gradient algorithm for minimizing linearly constrained separable convex problems and analyze its rate of convergence. In particular, we prove that under the assumption of strong convexity and Lipschitz continuity of the gradient of the primal objecti...
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| Veröffentlicht in: | Automatica (Oxford) Jg. 55; S. 209 - 216 |
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| Format: | Journal Article |
| Sprache: | Englisch |
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01.05.2015
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| Abstract | In this paper we propose a fully distributed dual gradient algorithm for minimizing linearly constrained separable convex problems and analyze its rate of convergence. In particular, we prove that under the assumption of strong convexity and Lipschitz continuity of the gradient of the primal objective function we have a global error bound type property for the dual problem. Using this error bound property we devise a fully distributed dual gradient scheme, i.e. a gradient scheme based on a weighted step size, for which we derive global linear rate of convergence for both dual and primal suboptimality and for primal feasibility violation. Numerical simulations are also provided to confirm our theory. |
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| AbstractList | In this paper we propose a fully distributed dual gradient algorithm for minimizing linearly constrained separable convex problems and analyze its rate of convergence. In particular, we prove that under the assumption of strong convexity and Lipschitz continuity of the gradient of the primal objective function we have a global error bound type property for the dual problem. Using this error bound property we devise a fully distributed dual gradient scheme, i.e. a gradient scheme based on a weighted step size, for which we derive global linear rate of convergence for both dual and primal suboptimality and for primal feasibility violation. Numerical simulations are also provided to confirm our theory. |
| Author | Necoara, Ion Nedelcu, Valentin |
| Author_xml | – sequence: 1 givenname: Ion surname: Necoara fullname: Necoara, Ion email: ion.necoara@acse.pub.ro – sequence: 2 givenname: Valentin surname: Nedelcu fullname: Nedelcu, Valentin email: valentin.nedelcu@acse.pub.ro |
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| Cites_doi | 10.1137/070708111 10.1007/s10589-013-9609-9 10.1109/TAC.2013.2275667 10.1109/TCNS.2014.2309751 10.1016/0024-3795(73)90007-4 10.1287/moor.18.4.846 10.1016/j.automatica.2013.01.009 10.1109/TPWRS.2003.814853 10.1109/TAC.2013.2294614 10.1016/j.jprocont.2010.12.010 10.1287/moor.12.3.474 |
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| References | Giselsson, Doan, Keviczky, De Schutter, Rantzer (br000020) 2013; 49 Luo, Tseng (br000030) 1993; 18 Necoara, Clipici (br000040) 2013 Pang (br000075) 1987; 12 Necoara, Nedelcu (br000050) 2014; 59 Hiriart-Urruty, Lemarechal (br000025) 1996 Nesterov (br000070) 2004 Beck, Nedic, Ozdaglar, Teboulle (br000010) 2014; 1 Necoara, Nedelcu, Dumitrache (br000055) 2011; 21 Necoara, Patrascu (br000060) 2014 Robinson (br000085) 1973; 6 Nedic, Ozdaglar (br000065) 2009; 19 Facchinei, Pang (br000015) 2003 Meinel, Ulbrich, Albrecht (br000035) 2014; 57 Patrinos, Bemporad (br000080) 2014; 59 Wang, Lin (br000090) 2013 Bakirtzis, Biskas (br000005) 2003; 18 Necoara, Nedelcu (br000045) 2013 Meinel (10.1016/j.automatica.2015.02.038_br000035) 2014; 57 Necoara (10.1016/j.automatica.2015.02.038_br000050) 2014; 59 Necoara (10.1016/j.automatica.2015.02.038_br000060) 2014 Giselsson (10.1016/j.automatica.2015.02.038_br000020) 2013; 49 Luo (10.1016/j.automatica.2015.02.038_br000030) 1993; 18 Hiriart-Urruty (10.1016/j.automatica.2015.02.038_br000025) 1996 Nesterov (10.1016/j.automatica.2015.02.038_br000070) 2004 Patrinos (10.1016/j.automatica.2015.02.038_br000080) 2014; 59 Beck (10.1016/j.automatica.2015.02.038_br000010) 2014; 1 Robinson (10.1016/j.automatica.2015.02.038_br000085) 1973; 6 Facchinei (10.1016/j.automatica.2015.02.038_br000015) 2003 Nedic (10.1016/j.automatica.2015.02.038_br000065) 2009; 19 Necoara (10.1016/j.automatica.2015.02.038_br000040) 2013 Bakirtzis (10.1016/j.automatica.2015.02.038_br000005) 2003; 18 Necoara (10.1016/j.automatica.2015.02.038_br000045) 2013 Wang (10.1016/j.automatica.2015.02.038_br000090) 2013 Necoara (10.1016/j.automatica.2015.02.038_br000055) 2011; 21 Pang (10.1016/j.automatica.2015.02.038_br000075) 1987; 12 |
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| SubjectTerms | Distributed gradient algorithm Dual decomposition Error bound Linear convergence Separable convex problems |
| Title | On linear convergence of a distributed dual gradient algorithm for linearly constrained separable convex problems |
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