A dynamical system method for solving the split convex feasibility problem

Zeng-Zhen Tan; Rong Hu; Ming Zhu; Ya-Ping Fang

doi:10.3934/jimo.2020104

Article Contents

2021, Volume 17, Issue 6: 2989-3011. Doi: 10.3934/jimo.2020104

This issue Previous Article Next Article Robust observer-based control for discrete-time semi-Markov jump systems with actuator saturation

A dynamical system method for solving the split convex feasibility problem

1.
Department of Mathematics, Sichuan University, Chengdu, Sichuan, China
2.
Department of Applied Mathematics, Chengdu University of Information Technology, Chengdu 610225, China
3.
Department of Mathematics, Sichuan University, Chengdu 610065, China

^* Corresponding author: Ya-Ping Fang
^* Corresponding author: Ya-Ping Fang

Received: August 2019

Revised: March 2020

Early access: June 2020

Published: November 2021

This work was partially supported by the National Science Foundation of China (No. 11471230) and the Scientific Research Foundation of the Education Department of Sichuan Province (No. 16ZA0213)

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Abstract

In this paper a dynamical system model is proposed for solving the split convex feasibility problem. Under mild conditions, it is shown that the proposed dynamical system globally converges to a solution of the split convex feasibility problem. An exponential convergence is obtained provided that the bounded linear regularity property is satisfied. The validity and transient behavior of the dynamical system is demonstrated by several numerical examples. The method proposed in this paper can be regarded as not only a continuous version but also an interior version of the known $ CQ $-method for solving the split convex feasibility problem.

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Mathematics Subject Classification: Primary: 65K05, 65K10, 90C25; Secondary: 47H09, 65L09.

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Figure 1. The transient behavior of dynamical system $ (4) $ with initial points $ x_0 = [1, 1, 1]^T $ in Example $ 1 $ via ode45

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Figure 2. The transient behavior of dynamical system $ (4) $ with initial points $ x_0 = [1, 2, 3]^T $ in Example $ 1 $ via the central difference method

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Figure 3. The transient behavior of dynamical system $ (4) $ with different $ \lambda $ in Example $ 1 $ via the explicit difference method

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Figure 4. The transient behavior of dynamical system $ (4) $ with initial points $ x_0 = [5, -4]^T $ in Example $ 2 $ via the explicit difference method

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Figure 5. The transient behavior of dynamical system $ (4) $ with initial points $ x_0 = [-10, 4]^T $ in Example $ 2 $ via the explicit difference method

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Figure 6. The transient behavior of dynamical system $ (29) $ with 20 random initial points in Example $ 3 $ via the explicit difference method

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Figure 7. The transient behavior of dynamical system $ (29) $ with $ x_0 = [-3, 5, 2]^T $ in Example $ 3 $ via the finite element method method

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Figure 8. The transient behavior of dynamical system $ (29) $ with $ x_0 = [-3, 5, 2]^T $ in Example $ 3 $ via the Piccard algorithm

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Figure 9. The transient behavior of dynamical system $ (4) $ with the initial point that is generated randomly in Example $ 4 $ via ode45

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Figure 10. The recovered sparse signal versus the true $ 50- $sparse signal in Example $ 4 $

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Figure 11. The objective function value against the time for the $ LASSO $ problem solved through the dynamical system $ (4) $ with different choices of $ \lambda $

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