Distributed Optimization in Prescribed-Time: Theory and Experiment
Xin Gong, Yukang Cui, Jun Shen, Junlin Xiong, Tingwen Huang
Abstract
This work studies the analysis and algorithm design problems of the distributed prescribed-time convex optimization of continuous-time multi-agent systems on undirected graphs. A new distributed algorithm with hybrid constant and time-specific update rates as well as predefined-time convergence guarantees is proposed, whose optimization period is independent of the initial states of all agents and the specific topology among agents. This algorithm consists of the following three cascading stages: 1) prescribed-time estimation of the average group gradients and Hessians over a time interval <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T_1$</tex-math></inline-formula> ; 2) prescribed-time consensus of all agents over a time interval <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T_2$</tex-math></inline-formula> ; 3) prescribed-time optimization to the global optimal point over a time interval <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T_3$</tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T_i>0$</tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$i=1\sim 3$</tex-math></inline-formula> , are three time intervals subject to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T=T_1+T_2+T_3$</tex-math></inline-formula> with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T$</tex-math></inline-formula> denoting a predefined time interval. Accordingly, it is possible to pre-assign the settling time arbitrarily according to task requirements. The validity of this new algorithm and the practicability of given theoretical results are illustrated via a simulation example and a prescribed-time optimal rendezvous formation experiment of an indoor unmanned aerial vehicle swarm.