Distributed Randomized Gradient-Free Convex Optimization With Set Constraints Over Time-Varying Weight-Unbalanced Digraphs
This paper explores a class of distributed constrained convex optimization problems where the objective function is a sum of <inline-formula><tex-math notation="LaTeX">N</tex-math></inline-formula> convex local objective functions. These functions are characterized...
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| Vydané v: | IEEE transactions on network science and engineering Ročník 12; číslo 2; s. 610 - 622 |
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| Hlavní autori: | , , , |
| Médium: | Journal Article |
| Jazyk: | English |
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IEEE
01.03.2025
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| ISSN: | 2327-4697, 2334-329X |
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| Abstract | This paper explores a class of distributed constrained convex optimization problems where the objective function is a sum of <inline-formula><tex-math notation="LaTeX">N</tex-math></inline-formula> convex local objective functions. These functions are characterized by local non-smoothness yet adhere to Lipschitz continuity, and the optimization process is further constrained by <inline-formula><tex-math notation="LaTeX">N</tex-math></inline-formula> distinct closed convex sets. To delineate the structure of information exchange among agents, a series of time-varying weight-unbalance directed graphs are introduced. Furthermore, this study introduces a novel algorithm, distributed randomized gradient-free constrained optimization algorithm. This algorithm marks a significant advancement by substituting the conventional requirement for precise gradient or subgradient information in each iterative update with a random gradient-free oracle, thereby addressing scenarios where accurate gradient information is hard to obtain. A thorough convergence analysis is provided based on the smoothing parameters inherent in the local objective functions, the Lipschitz constants, and a series of standard assumptions. Significantly, the proposed algorithm can converge to an approximate optimal solution within a predetermined error threshold for the consisdered optimization problem, achieving the same convergence rate of <inline-formula><tex-math notation="LaTeX">{\mathcal O}(\frac{\ln (k)}{\sqrt{k} })</tex-math></inline-formula> as the general randomized gradient-free algorithms when the decay step size is selected appropriately. And when at least one of the local objective functions exhibits strong convexity, the proposed algorithm can achieve a faster convergence rate, <inline-formula><tex-math notation="LaTeX">{\mathcal O}(\frac{1}{k})</tex-math></inline-formula>. Finally, rigorous simulation results verify the correctness of theoretical findings. |
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| AbstractList | This paper explores a class of distributed constrained convex optimization problems where the objective function is a sum of <inline-formula><tex-math notation="LaTeX">N</tex-math></inline-formula> convex local objective functions. These functions are characterized by local non-smoothness yet adhere to Lipschitz continuity, and the optimization process is further constrained by <inline-formula><tex-math notation="LaTeX">N</tex-math></inline-formula> distinct closed convex sets. To delineate the structure of information exchange among agents, a series of time-varying weight-unbalance directed graphs are introduced. Furthermore, this study introduces a novel algorithm, distributed randomized gradient-free constrained optimization algorithm. This algorithm marks a significant advancement by substituting the conventional requirement for precise gradient or subgradient information in each iterative update with a random gradient-free oracle, thereby addressing scenarios where accurate gradient information is hard to obtain. A thorough convergence analysis is provided based on the smoothing parameters inherent in the local objective functions, the Lipschitz constants, and a series of standard assumptions. Significantly, the proposed algorithm can converge to an approximate optimal solution within a predetermined error threshold for the consisdered optimization problem, achieving the same convergence rate of <inline-formula><tex-math notation="LaTeX">{\mathcal O}(\frac{\ln (k)}{\sqrt{k} })</tex-math></inline-formula> as the general randomized gradient-free algorithms when the decay step size is selected appropriately. And when at least one of the local objective functions exhibits strong convexity, the proposed algorithm can achieve a faster convergence rate, <inline-formula><tex-math notation="LaTeX">{\mathcal O}(\frac{1}{k})</tex-math></inline-formula>. Finally, rigorous simulation results verify the correctness of theoretical findings. This paper explores a class of distributed constrained convex optimization problems where the objective function is a sum of [Formula Omitted] convex local objective functions. These functions are characterized by local non-smoothness yet adhere to Lipschitz continuity, and the optimization process is further constrained by [Formula Omitted] distinct closed convex sets. To delineate the structure of information exchange among agents, a series of time-varying weight-unbalance directed graphs are introduced. Furthermore, this study introduces a novel algorithm, distributed randomized gradient-free constrained optimization algorithm. This algorithm marks a significant advancement by substituting the conventional requirement for precise gradient or subgradient information in each iterative update with a random gradient-free oracle, thereby addressing scenarios where accurate gradient information is hard to obtain. A thorough convergence analysis is provided based on the smoothing parameters inherent in the local objective functions, the Lipschitz constants, and a series of standard assumptions. Significantly, the proposed algorithm can converge to an approximate optimal solution within a predetermined error threshold for the consisdered optimization problem, achieving the same convergence rate of [Formula Omitted] as the general randomized gradient-free algorithms when the decay step size is selected appropriately. And when at least one of the local objective functions exhibits strong convexity, the proposed algorithm can achieve a faster convergence rate, [Formula Omitted]. Finally, rigorous simulation results verify the correctness of theoretical findings. |
| Author | Zhu, Yanan Li, Tao Wen, Guanghui Li, Qinghai |
| Author_xml | – sequence: 1 givenname: Yanan orcidid: 0000-0001-7980-6282 surname: Zhu fullname: Zhu, Yanan email: ynzhu@nuist.edu.cn organization: School of Automation, Nanjing University of Information Science and Technology, Nanjing, China – sequence: 2 givenname: Qinghai surname: Li fullname: Li, Qinghai email: 202212490051@nuist.edu.cn organization: School of Automation, Nanjing University of Information Science and Technology, Nanjing, China – sequence: 3 givenname: Tao orcidid: 0000-0002-9190-0199 surname: Li fullname: Li, Tao email: litaojia@nuist.edu.cn organization: School of Automation, Nanjing University of Information Science and Technology, Nanjing, China – sequence: 4 givenname: Guanghui orcidid: 0000-0003-0070-8597 surname: Wen fullname: Wen, Guanghui email: wenguanghui@gmail.com organization: Department of Systems Science, School of Mathematics, Southeast University, Nanjing, China |
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| SubjectTerms | Algorithms Closed box Constraints Convergence Convergence rate Convex analysis Convex functions Convexity Directed graphs distributed multi-agent optimization Graph theory Linear programming Lipschitz condition Optimization Prediction algorithms randomized gradient-free Smoothing methods Smoothness Stochastic processes time-varying weight-unbalanced digraphs Vectors |
| Title | Distributed Randomized Gradient-Free Convex Optimization With Set Constraints Over Time-Varying Weight-Unbalanced Digraphs |
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