Frequency-Domain Analysis of Distributed Optimization: Fundamental Convergence Rate and Optimal Algorithm Synthesis

The design of optimization algorithms has long been a matter of art, and it calls for the systematic development of methods in which algorithms can be analyzed, designed, and benchmarked with regard to their key attributes such as efficiency, complexity, and robustness. This article answers this nee...

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Vydáno v:IEEE transactions on automatic control Ročník 69; číslo 12; s. 8539 - 8554
Hlavní autoři: Zhang, Shiqi, Wu, Wuwei, Li, Zhongkui, Chen, Jie, Georgiou, Tryphon T.
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York IEEE 01.12.2024
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Témata:
Algorithms
Circle criterion
Complexity
Convergence
Design optimization
Distributed algorithms
Distributed optimization
Frequency analysis
Frequency domain analysis
frequency-domain methods
gradient-based algorithms
Heuristic algorithms
Interpolation
Nevanlinna–Pick interpolation
Nonlinear control
Optimal control
Optimization
robust and <named-content xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" content-type="math" xlink:type="simple"> <inline-formula> <tex-math notation="LaTeX"> mathcal {H}_{\infty }</tex-math> </inline-formula> </named-content> control
Robust control
Stability criteria
Subsystems
Synthesis
ISSN:0018-9286, 1558-2523
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Shrnutí:The design of optimization algorithms has long been a matter of art, and it calls for the systematic development of methods in which algorithms can be analyzed, designed, and benchmarked with regard to their key attributes such as efficiency, complexity, and robustness. This article answers this need by providing a frequency-domain framework for algorithm analysis and synthesis, with a particular emphasis on distributed optimization problems. We propose a general class of gradient-based distributed algorithms that can be viewed as interconnected dynamical systems consisting of a linear time-invariant subsystem and a Lur'e-type nonlinear component, thereby enabling analysis and synthesis of such algorithms from a robust control perspective via the usage of the circle criterion and the Zames-Falb theorem. By linking the convergence of an algorithm with the absolute stability of a corresponding Lur'e system, optimization of the algorithmic convergence rate is recast as a Nevanlinna-Pick interpolation problem akin to an <inline-formula><tex-math notation="LaTeX">\mathcal {H}_{\infty }</tex-math></inline-formula> optimal control problem. Solutions to such interpolation problems lead to a variety of gradient-based algorithms with optimal and suboptimal convergence rates, and algorithmic complexity judiciously controlled.
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ISSN:0018-9286
1558-2523
DOI:10.1109/TAC.2024.3413854