[Paper Review] Finite-Time Distributed Linear Equation Solver for Minimum $l_1$ Norm Solutions

Jingqiu Zhou, Xuan Wang+2 more|arXiv (Cornell University)|Sep 28, 2017

Matrix Theory and Algorithms32 references4 citations

TL;DR

This paper proposes a finite-time distributed algorithm for multi-agent networks to solve linear equations $Ax = b$ with the minimum $l_1$-norm solution in the underdetermined case. By combining projection-consensus flow with finite-time gradient dynamics, the method ensures all agents achieve consensus on the minimum $l_1$-norm solution in finite time, proven via Filippov differential inclusions and generalized Lie derivatives, validated by simulations on a four-agent network.

ABSTRACT

This paper proposes distributed algorithms for multi-agent networks to achieve a solution in finite time to a linear equation $Ax=b$ where $A$ has full row rank, and with the minimum $l_1$-norm in the underdetermined case (where $A$ has more columns than rows). The underlying network is assumed to be undirected and fixed, and an analytical proof is provided for the proposed algorithm to drive all agents' individual states to converge to a common value, viz a solution of $Ax=b$, which is the minimum $l_1$-norm solution in the underdetermined case. Numerical simulations are also provided as validation of the proposed algorithms.

Motivation & Objective

• To develop a distributed algorithm that achieves finite-time convergence to a solution of $Ax = b$ in multi-agent networks.
• To extend the finite-time convergence property to the minimum $l_1$-norm solution in the underdetermined case, where $A$ has more columns than rows.
• To eliminate asymptotic convergence limitations of existing methods by ensuring exact convergence in finite time.
• To provide analytical proof of finite-time convergence using Filippov theory and generalized Lie derivatives.
• To validate the algorithm through numerical simulations on a four-agent network with varying convergence dynamics.

Proposed method

• The algorithm uses a distributed update rule based on projection-consensus flow, enabling each agent to estimate the solution using local information and neighbor communication.
• Finite-time convergence is achieved by incorporating a time-varying gain $k(t) = \frac{\bar{\delta}}{t+1} + \delta$ in the update law, ensuring rapid convergence.
• The method leverages Filippov set-valued maps and generalized Lie derivatives to analyze differential inclusions arising from the non-smooth $\text{sgn}(\cdot)$ function in the dynamics.
• A centralized update for minimum $l_1$-norm solutions is first derived using a finite-time gradient flow on the $l_1$-norm objective, then embedded into the distributed framework.
• The overall distributed algorithm combines the finite-time consensus mechanism with the centralized minimum $l_1$-norm solver, ensuring all agents converge to the same minimum $l_1$-norm solution.
• The convergence is validated using the $l_1$-norm of the deviation from consensus, $\|y(t) - \mathbf{1}_m \otimes x^*\|_1$, which reaches zero in finite time.

Experimental results

Research questions

• RQ1 Can a distributed algorithm achieve finite-time convergence to a solution of $Ax = b$ in a multi-agent network with only local communication?
• RQ2 How can the minimum $l_1$-norm solution be achieved in finite time in an underdetermined system using distributed computation?
• RQ3 What dynamical mechanisms ensure finite-time convergence despite the non-smoothness of the $l_1$-norm and $\text{sgn}(\cdot)$ functions?
• RQ4 How does the time-varying gain $k(t)$ influence the convergence rate and finite-time behavior of the distributed solver?
• RQ5 What is the relationship between the convergence time of the distributed algorithm and the centralized minimum $l_1$-norm solver?

Key findings

• The distributed algorithm achieves finite-time convergence to a solution of $Ax = b$ in a four-agent network, as evidenced by $\|y(t) - \mathbf{1}_m \otimes x^*\|_1$ reaching zero in finite time.
• The centralized update for minimum $l_1$-norm solutions achieves finite-time convergence, with $\|y(t) - \bar{x}^*\|_1$ reaching zero and remaining zero.
• The distributed algorithm for minimum $l_1$-norm solutions converges in finite time regardless of different parameter choices for $\bar{\delta}$ and $\delta$, with convergence time decreasing as $\delta$ increases.
• Increasing $\delta$ leads to significantly faster convergence, while increasing $\bar{\delta}$ also improves convergence, though less dramatically.
• The convergence time of the distributed algorithm is approximately $\frac{1}{\delta}$ times longer than the centralized case, due to the scaling of the centralized dynamics by $k(t) < 1$.
• Consensus among agents is achieved in finite time, as shown by $\|y(t) - \mathbf{1}_m \otimes \bar{y}(t)\|_1$ reaching zero, indicating synchronized convergence to the same solution.