A fixed step distributed proximal gradient push‐pull algorithm based on integral quadratic constraint
In order to solve the distributed optimization problem with smooth + nonsmooth structure of the objective function on unbalanced directed networks, this article uses the proximal operator to deal with the nonsmooth part of the objective function, and designs and analyzes the fixed step proximal grad...
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| Published in: | Optimal control applications & methods Vol. 44; no. 5; pp. 2693 - 2707 |
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| Format: | Journal Article |
| Language: | English |
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01.09.2023
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| ISSN: | 0143-2087, 1099-1514 |
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| Abstract | In order to solve the distributed optimization problem with smooth + nonsmooth structure of the objective function on unbalanced directed networks, this article uses the proximal operator to deal with the nonsmooth part of the objective function, and designs and analyzes the fixed step proximal gradient Push‐Pull (PG‐Push‐Pull) algorithm. Firstly, the Integral Quadratic Constraint (IQC) suitable for proximal gradient Push‐Pull algorithm is given. When the smooth part of the objective function is strongly convex and the gradient satisfies the Lipchitz condition, the convergence of the algorithm is proved, and the convergence analysis is transformed into solving a linear matrix inequality by using this IQC framework. Its feasibility can ensure that the proposed algorithm has linear convergence rate, which is the same as that of Push‐Pull gradient algorithm. Then, the upper bound of convergence rate can be found by solving a Non‐Linear Programming problem. Finally, an example is given to analyze the upper bound of the convergence rate and verify the effectiveness of the proposed algorithm. |
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| AbstractList | In order to solve the distributed optimization problem with smooth + nonsmooth structure of the objective function on unbalanced directed networks, this article uses the proximal operator to deal with the nonsmooth part of the objective function, and designs and analyzes the fixed step proximal gradient Push‐Pull (PG‐Push‐Pull) algorithm. Firstly, the Integral Quadratic Constraint (IQC) suitable for proximal gradient Push‐Pull algorithm is given. When the smooth part of the objective function is strongly convex and the gradient satisfies the Lipchitz condition, the convergence of the algorithm is proved, and the convergence analysis is transformed into solving a linear matrix inequality by using this IQC framework. Its feasibility can ensure that the proposed algorithm has linear convergence rate, which is the same as that of Push‐Pull gradient algorithm. Then, the upper bound of convergence rate can be found by solving a Non‐Linear Programming problem. Finally, an example is given to analyze the upper bound of the convergence rate and verify the effectiveness of the proposed algorithm. In order to solve the distributed optimization problem with smooth + nonsmooth structure of the objective function on unbalanced directed networks, this article uses the proximal operator to deal with the nonsmooth part of the objective function, and designs and analyzes the fixed step proximal gradient Push‐Pull (PG‐Push‐Pull) algorithm. Firstly, the Integral Quadratic Constraint (IQC) suitable for proximal gradient Push‐Pull algorithm is given. When the smooth part of the objective function is strongly convex and the gradient satisfies the Lipchitz condition, the convergence of the algorithm is proved, and the convergence analysis is transformed into solving a linear matrix inequality by using this IQC framework. Its feasibility can ensure that the proposed algorithm has linear convergence rate, which is the same as that of Push‐Pull gradient algorithm. Then, the upper bound of convergence rate can be found by solving a Non‐Linear Programming problem. Finally, an example is given to analyze the upper bound of the convergence rate and verify the effectiveness of the proposed algorithm. |
| Author | Xie, Yibin Gao, Wenhua Ren, Hongwei |
| Author_xml | – sequence: 1 givenname: Wenhua orcidid: 0000-0002-6439-1915 surname: Gao fullname: Gao, Wenhua organization: School of Mathematics South China University of Technology Guangzhou China – sequence: 2 givenname: Yibin surname: Xie fullname: Xie, Yibin organization: School of Mathematics South China University of Technology Guangzhou China – sequence: 3 givenname: Hongwei surname: Ren fullname: Ren, Hongwei organization: School of Automation Guangdong University of Petrochemical Technology Maoming China |
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| Cites_doi | 10.1109/TAC.2008.2009515 10.1002/oca.2254 10.1137/17M1136845 10.1109/TCNS.2020.2988009 10.1109/ICCA.2019.8899565 10.1137/14096668X 10.1137/16M1084316 10.1109/TSMC.2017.2757265 10.1002/oca.2534 10.1002/oca.2136 10.1515/9781400873173 10.1109/ALLERTON.2017.8262874 10.1109/GLOCOM.2013.6831603 10.1109/9.587335 10.1016/j.ifacol.2019.12.181 10.1109/TAC.2020.2972824 10.1109/TSMC.2019.2933005 10.1109/TSP.2015.2461520 10.1109/TCNS.2017.2698261 10.1109/TAC.2014.2364096 10.1109/MSP.2020.2975210 10.1109/TIT.2015.2403263 10.1002/oca.2424 10.1109/TCYB.2018.2890140 10.1137/15M1009597 10.23919/ACC.2019.8815081 |
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| Snippet | In order to solve the distributed optimization problem with smooth + nonsmooth structure of the objective function on unbalanced directed networks, this... In order to solve the distributed optimization problem with smooth + nonsmooth structure of the objective function on unbalanced directed networks, this... |
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| SubjectTerms | Algorithms Convergence Linear matrix inequalities Linear programming Mathematical analysis Operators (mathematics) Optimization Upper bounds |
| Title | A fixed step distributed proximal gradient push‐pull algorithm based on integral quadratic constraint |
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