On the Sublinear Convergence Rate of Multi-block ADMM
Abstract The alternating direction method of multipliers (ADMM) is widely used in solving structured convex optimization problems. Despite its success in practice, the convergence of the standard ADMM for minimizing the sum of %$N\,(N\geqslant 3)%$ convex functions, whose variables are linked by lin...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Lin, Tian-Yi [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2015 |
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| Schlagwörter: |
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| Anmerkung: |
© Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag Berlin Heidelberg 2015 |
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| Übergeordnetes Werk: |
Enthalten in: Journal of the operations research society of China - Berlin : Springer, 2013, 3(2015), 3 vom: 19. Juli, Seite 251-274 |
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| Übergeordnetes Werk: |
volume:3 ; year:2015 ; number:3 ; day:19 ; month:07 ; pages:251-274 |
| Links: |
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| DOI / URN: |
10.1007/s40305-015-0092-0 |
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| Katalog-ID: |
SPR036575704 |
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| 245 | 1 | 0 | |a On the Sublinear Convergence Rate of Multi-block ADMM |
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| 520 | |a Abstract The alternating direction method of multipliers (ADMM) is widely used in solving structured convex optimization problems. Despite its success in practice, the convergence of the standard ADMM for minimizing the sum of %$N\,(N\geqslant 3)%$ convex functions, whose variables are linked by linear constraints, has remained unclear for a very long time. Recently, Chen et al. (Math Program, doi:10.1007/s10107-014-0826-5, 2014) provided a counter-example showing that the ADMM for %$N\geqslant 3%$ may fail to converge without further conditions. Since the ADMM for %$N\geqslant 3%$ has been very successful when applied to many problems arising from real practice, it is worth further investigating under what kind of sufficient conditions it can be guaranteed to converge. In this paper, we present such sufficient conditions that can guarantee the sublinear convergence rate for the ADMM for %$N\geqslant 3%$. Specifically, we show that if one of the functions is convex (not necessarily strongly convex) and the other N-1 functions are strongly convex, and the penalty parameter lies in a certain region, the ADMM converges with rate O(1 / t) in a certain ergodic sense and o(1 / t) in a certain non-ergodic sense, where t denotes the number of iterations. As a by-product, we also provide a simple proof for the O(1 / t) convergence rate of two-block ADMM in terms of both objective error and constraint violation, without assuming any condition on the penalty parameter and strong convexity on the functions. | ||
| 650 | 4 | |a Alternating direction method of multipliers |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Sublinear convergence rate |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Convex optimization |7 (dpeaa)DE-He213 | |
| 700 | 1 | |a Ma, Shi-Qian |4 aut | |
| 700 | 1 | |a Zhang, Shu-Zhong |4 aut | |
| 773 | 0 | 8 | |i Enthalten in |t Journal of the operations research society of China |d Berlin : Springer, 2013 |g 3(2015), 3 vom: 19. Juli, Seite 251-274 |w (DE-627)739215051 |w (DE-600)2708006-7 |x 2194-6698 |7 nnns |
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10.1007/s40305-015-0092-0 doi (DE-627)SPR036575704 (SPR)s40305-015-0092-0-e DE-627 ger DE-627 rakwb eng Lin, Tian-Yi verfasserin aut On the Sublinear Convergence Rate of Multi-block ADMM 2015 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag Berlin Heidelberg 2015 Abstract The alternating direction method of multipliers (ADMM) is widely used in solving structured convex optimization problems. Despite its success in practice, the convergence of the standard ADMM for minimizing the sum of %$N\,(N\geqslant 3)%$ convex functions, whose variables are linked by linear constraints, has remained unclear for a very long time. Recently, Chen et al. (Math Program, doi:10.1007/s10107-014-0826-5, 2014) provided a counter-example showing that the ADMM for %$N\geqslant 3%$ may fail to converge without further conditions. Since the ADMM for %$N\geqslant 3%$ has been very successful when applied to many problems arising from real practice, it is worth further investigating under what kind of sufficient conditions it can be guaranteed to converge. In this paper, we present such sufficient conditions that can guarantee the sublinear convergence rate for the ADMM for %$N\geqslant 3%$. Specifically, we show that if one of the functions is convex (not necessarily strongly convex) and the other N-1 functions are strongly convex, and the penalty parameter lies in a certain region, the ADMM converges with rate O(1 / t) in a certain ergodic sense and o(1 / t) in a certain non-ergodic sense, where t denotes the number of iterations. As a by-product, we also provide a simple proof for the O(1 / t) convergence rate of two-block ADMM in terms of both objective error and constraint violation, without assuming any condition on the penalty parameter and strong convexity on the functions. Alternating direction method of multipliers (dpeaa)DE-He213 Sublinear convergence rate (dpeaa)DE-He213 Convex optimization (dpeaa)DE-He213 Ma, Shi-Qian aut Zhang, Shu-Zhong aut Enthalten in Journal of the operations research society of China Berlin : Springer, 2013 3(2015), 3 vom: 19. Juli, Seite 251-274 (DE-627)739215051 (DE-600)2708006-7 2194-6698 nnns volume:3 year:2015 number:3 day:19 month:07 pages:251-274 https://dx.doi.org/10.1007/s40305-015-0092-0 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 3 2015 3 19 07 251-274 |
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10.1007/s40305-015-0092-0 doi (DE-627)SPR036575704 (SPR)s40305-015-0092-0-e DE-627 ger DE-627 rakwb eng Lin, Tian-Yi verfasserin aut On the Sublinear Convergence Rate of Multi-block ADMM 2015 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag Berlin Heidelberg 2015 Abstract The alternating direction method of multipliers (ADMM) is widely used in solving structured convex optimization problems. Despite its success in practice, the convergence of the standard ADMM for minimizing the sum of %$N\,(N\geqslant 3)%$ convex functions, whose variables are linked by linear constraints, has remained unclear for a very long time. Recently, Chen et al. (Math Program, doi:10.1007/s10107-014-0826-5, 2014) provided a counter-example showing that the ADMM for %$N\geqslant 3%$ may fail to converge without further conditions. Since the ADMM for %$N\geqslant 3%$ has been very successful when applied to many problems arising from real practice, it is worth further investigating under what kind of sufficient conditions it can be guaranteed to converge. In this paper, we present such sufficient conditions that can guarantee the sublinear convergence rate for the ADMM for %$N\geqslant 3%$. Specifically, we show that if one of the functions is convex (not necessarily strongly convex) and the other N-1 functions are strongly convex, and the penalty parameter lies in a certain region, the ADMM converges with rate O(1 / t) in a certain ergodic sense and o(1 / t) in a certain non-ergodic sense, where t denotes the number of iterations. As a by-product, we also provide a simple proof for the O(1 / t) convergence rate of two-block ADMM in terms of both objective error and constraint violation, without assuming any condition on the penalty parameter and strong convexity on the functions. Alternating direction method of multipliers (dpeaa)DE-He213 Sublinear convergence rate (dpeaa)DE-He213 Convex optimization (dpeaa)DE-He213 Ma, Shi-Qian aut Zhang, Shu-Zhong aut Enthalten in Journal of the operations research society of China Berlin : Springer, 2013 3(2015), 3 vom: 19. Juli, Seite 251-274 (DE-627)739215051 (DE-600)2708006-7 2194-6698 nnns volume:3 year:2015 number:3 day:19 month:07 pages:251-274 https://dx.doi.org/10.1007/s40305-015-0092-0 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 3 2015 3 19 07 251-274 |
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10.1007/s40305-015-0092-0 doi (DE-627)SPR036575704 (SPR)s40305-015-0092-0-e DE-627 ger DE-627 rakwb eng Lin, Tian-Yi verfasserin aut On the Sublinear Convergence Rate of Multi-block ADMM 2015 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag Berlin Heidelberg 2015 Abstract The alternating direction method of multipliers (ADMM) is widely used in solving structured convex optimization problems. Despite its success in practice, the convergence of the standard ADMM for minimizing the sum of %$N\,(N\geqslant 3)%$ convex functions, whose variables are linked by linear constraints, has remained unclear for a very long time. Recently, Chen et al. (Math Program, doi:10.1007/s10107-014-0826-5, 2014) provided a counter-example showing that the ADMM for %$N\geqslant 3%$ may fail to converge without further conditions. Since the ADMM for %$N\geqslant 3%$ has been very successful when applied to many problems arising from real practice, it is worth further investigating under what kind of sufficient conditions it can be guaranteed to converge. In this paper, we present such sufficient conditions that can guarantee the sublinear convergence rate for the ADMM for %$N\geqslant 3%$. Specifically, we show that if one of the functions is convex (not necessarily strongly convex) and the other N-1 functions are strongly convex, and the penalty parameter lies in a certain region, the ADMM converges with rate O(1 / t) in a certain ergodic sense and o(1 / t) in a certain non-ergodic sense, where t denotes the number of iterations. As a by-product, we also provide a simple proof for the O(1 / t) convergence rate of two-block ADMM in terms of both objective error and constraint violation, without assuming any condition on the penalty parameter and strong convexity on the functions. Alternating direction method of multipliers (dpeaa)DE-He213 Sublinear convergence rate (dpeaa)DE-He213 Convex optimization (dpeaa)DE-He213 Ma, Shi-Qian aut Zhang, Shu-Zhong aut Enthalten in Journal of the operations research society of China Berlin : Springer, 2013 3(2015), 3 vom: 19. Juli, Seite 251-274 (DE-627)739215051 (DE-600)2708006-7 2194-6698 nnns volume:3 year:2015 number:3 day:19 month:07 pages:251-274 https://dx.doi.org/10.1007/s40305-015-0092-0 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 3 2015 3 19 07 251-274 |
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10.1007/s40305-015-0092-0 doi (DE-627)SPR036575704 (SPR)s40305-015-0092-0-e DE-627 ger DE-627 rakwb eng Lin, Tian-Yi verfasserin aut On the Sublinear Convergence Rate of Multi-block ADMM 2015 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag Berlin Heidelberg 2015 Abstract The alternating direction method of multipliers (ADMM) is widely used in solving structured convex optimization problems. Despite its success in practice, the convergence of the standard ADMM for minimizing the sum of %$N\,(N\geqslant 3)%$ convex functions, whose variables are linked by linear constraints, has remained unclear for a very long time. Recently, Chen et al. (Math Program, doi:10.1007/s10107-014-0826-5, 2014) provided a counter-example showing that the ADMM for %$N\geqslant 3%$ may fail to converge without further conditions. Since the ADMM for %$N\geqslant 3%$ has been very successful when applied to many problems arising from real practice, it is worth further investigating under what kind of sufficient conditions it can be guaranteed to converge. In this paper, we present such sufficient conditions that can guarantee the sublinear convergence rate for the ADMM for %$N\geqslant 3%$. Specifically, we show that if one of the functions is convex (not necessarily strongly convex) and the other N-1 functions are strongly convex, and the penalty parameter lies in a certain region, the ADMM converges with rate O(1 / t) in a certain ergodic sense and o(1 / t) in a certain non-ergodic sense, where t denotes the number of iterations. As a by-product, we also provide a simple proof for the O(1 / t) convergence rate of two-block ADMM in terms of both objective error and constraint violation, without assuming any condition on the penalty parameter and strong convexity on the functions. Alternating direction method of multipliers (dpeaa)DE-He213 Sublinear convergence rate (dpeaa)DE-He213 Convex optimization (dpeaa)DE-He213 Ma, Shi-Qian aut Zhang, Shu-Zhong aut Enthalten in Journal of the operations research society of China Berlin : Springer, 2013 3(2015), 3 vom: 19. Juli, Seite 251-274 (DE-627)739215051 (DE-600)2708006-7 2194-6698 nnns volume:3 year:2015 number:3 day:19 month:07 pages:251-274 https://dx.doi.org/10.1007/s40305-015-0092-0 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 3 2015 3 19 07 251-274 |
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10.1007/s40305-015-0092-0 doi (DE-627)SPR036575704 (SPR)s40305-015-0092-0-e DE-627 ger DE-627 rakwb eng Lin, Tian-Yi verfasserin aut On the Sublinear Convergence Rate of Multi-block ADMM 2015 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag Berlin Heidelberg 2015 Abstract The alternating direction method of multipliers (ADMM) is widely used in solving structured convex optimization problems. Despite its success in practice, the convergence of the standard ADMM for minimizing the sum of %$N\,(N\geqslant 3)%$ convex functions, whose variables are linked by linear constraints, has remained unclear for a very long time. Recently, Chen et al. (Math Program, doi:10.1007/s10107-014-0826-5, 2014) provided a counter-example showing that the ADMM for %$N\geqslant 3%$ may fail to converge without further conditions. Since the ADMM for %$N\geqslant 3%$ has been very successful when applied to many problems arising from real practice, it is worth further investigating under what kind of sufficient conditions it can be guaranteed to converge. In this paper, we present such sufficient conditions that can guarantee the sublinear convergence rate for the ADMM for %$N\geqslant 3%$. Specifically, we show that if one of the functions is convex (not necessarily strongly convex) and the other N-1 functions are strongly convex, and the penalty parameter lies in a certain region, the ADMM converges with rate O(1 / t) in a certain ergodic sense and o(1 / t) in a certain non-ergodic sense, where t denotes the number of iterations. As a by-product, we also provide a simple proof for the O(1 / t) convergence rate of two-block ADMM in terms of both objective error and constraint violation, without assuming any condition on the penalty parameter and strong convexity on the functions. Alternating direction method of multipliers (dpeaa)DE-He213 Sublinear convergence rate (dpeaa)DE-He213 Convex optimization (dpeaa)DE-He213 Ma, Shi-Qian aut Zhang, Shu-Zhong aut Enthalten in Journal of the operations research society of China Berlin : Springer, 2013 3(2015), 3 vom: 19. Juli, Seite 251-274 (DE-627)739215051 (DE-600)2708006-7 2194-6698 nnns volume:3 year:2015 number:3 day:19 month:07 pages:251-274 https://dx.doi.org/10.1007/s40305-015-0092-0 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 3 2015 3 19 07 251-274 |
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Despite its success in practice, the convergence of the standard ADMM for minimizing the sum of %$N\,(N\geqslant 3)%$ convex functions, whose variables are linked by linear constraints, has remained unclear for a very long time. Recently, Chen et al. (Math Program, doi:10.1007/s10107-014-0826-5, 2014) provided a counter-example showing that the ADMM for %$N\geqslant 3%$ may fail to converge without further conditions. Since the ADMM for %$N\geqslant 3%$ has been very successful when applied to many problems arising from real practice, it is worth further investigating under what kind of sufficient conditions it can be guaranteed to converge. In this paper, we present such sufficient conditions that can guarantee the sublinear convergence rate for the ADMM for %$N\geqslant 3%$. Specifically, we show that if one of the functions is convex (not necessarily strongly convex) and the other N-1 functions are strongly convex, and the penalty parameter lies in a certain region, the ADMM converges with rate O(1 / t) in a certain ergodic sense and o(1 / t) in a certain non-ergodic sense, where t denotes the number of iterations. 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Lin, Tian-Yi |
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On the Sublinear Convergence Rate of Multi-block ADMM Alternating direction method of multipliers (dpeaa)DE-He213 Sublinear convergence rate (dpeaa)DE-He213 Convex optimization (dpeaa)DE-He213 |
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On the Sublinear Convergence Rate of Multi-block ADMM |
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on the sublinear convergence rate of multi-block admm |
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On the Sublinear Convergence Rate of Multi-block ADMM |
| abstract |
Abstract The alternating direction method of multipliers (ADMM) is widely used in solving structured convex optimization problems. Despite its success in practice, the convergence of the standard ADMM for minimizing the sum of %$N\,(N\geqslant 3)%$ convex functions, whose variables are linked by linear constraints, has remained unclear for a very long time. Recently, Chen et al. (Math Program, doi:10.1007/s10107-014-0826-5, 2014) provided a counter-example showing that the ADMM for %$N\geqslant 3%$ may fail to converge without further conditions. Since the ADMM for %$N\geqslant 3%$ has been very successful when applied to many problems arising from real practice, it is worth further investigating under what kind of sufficient conditions it can be guaranteed to converge. In this paper, we present such sufficient conditions that can guarantee the sublinear convergence rate for the ADMM for %$N\geqslant 3%$. Specifically, we show that if one of the functions is convex (not necessarily strongly convex) and the other N-1 functions are strongly convex, and the penalty parameter lies in a certain region, the ADMM converges with rate O(1 / t) in a certain ergodic sense and o(1 / t) in a certain non-ergodic sense, where t denotes the number of iterations. As a by-product, we also provide a simple proof for the O(1 / t) convergence rate of two-block ADMM in terms of both objective error and constraint violation, without assuming any condition on the penalty parameter and strong convexity on the functions. © Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag Berlin Heidelberg 2015 |
| abstractGer |
Abstract The alternating direction method of multipliers (ADMM) is widely used in solving structured convex optimization problems. Despite its success in practice, the convergence of the standard ADMM for minimizing the sum of %$N\,(N\geqslant 3)%$ convex functions, whose variables are linked by linear constraints, has remained unclear for a very long time. Recently, Chen et al. (Math Program, doi:10.1007/s10107-014-0826-5, 2014) provided a counter-example showing that the ADMM for %$N\geqslant 3%$ may fail to converge without further conditions. Since the ADMM for %$N\geqslant 3%$ has been very successful when applied to many problems arising from real practice, it is worth further investigating under what kind of sufficient conditions it can be guaranteed to converge. In this paper, we present such sufficient conditions that can guarantee the sublinear convergence rate for the ADMM for %$N\geqslant 3%$. Specifically, we show that if one of the functions is convex (not necessarily strongly convex) and the other N-1 functions are strongly convex, and the penalty parameter lies in a certain region, the ADMM converges with rate O(1 / t) in a certain ergodic sense and o(1 / t) in a certain non-ergodic sense, where t denotes the number of iterations. As a by-product, we also provide a simple proof for the O(1 / t) convergence rate of two-block ADMM in terms of both objective error and constraint violation, without assuming any condition on the penalty parameter and strong convexity on the functions. © Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag Berlin Heidelberg 2015 |
| abstract_unstemmed |
Abstract The alternating direction method of multipliers (ADMM) is widely used in solving structured convex optimization problems. Despite its success in practice, the convergence of the standard ADMM for minimizing the sum of %$N\,(N\geqslant 3)%$ convex functions, whose variables are linked by linear constraints, has remained unclear for a very long time. Recently, Chen et al. (Math Program, doi:10.1007/s10107-014-0826-5, 2014) provided a counter-example showing that the ADMM for %$N\geqslant 3%$ may fail to converge without further conditions. Since the ADMM for %$N\geqslant 3%$ has been very successful when applied to many problems arising from real practice, it is worth further investigating under what kind of sufficient conditions it can be guaranteed to converge. In this paper, we present such sufficient conditions that can guarantee the sublinear convergence rate for the ADMM for %$N\geqslant 3%$. Specifically, we show that if one of the functions is convex (not necessarily strongly convex) and the other N-1 functions are strongly convex, and the penalty parameter lies in a certain region, the ADMM converges with rate O(1 / t) in a certain ergodic sense and o(1 / t) in a certain non-ergodic sense, where t denotes the number of iterations. As a by-product, we also provide a simple proof for the O(1 / t) convergence rate of two-block ADMM in terms of both objective error and constraint violation, without assuming any condition on the penalty parameter and strong convexity on the functions. © Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag Berlin Heidelberg 2015 |
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On the Sublinear Convergence Rate of Multi-block ADMM |
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Ma, Shi-Qian Zhang, Shu-Zhong |
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