4 research outputs found
Finite-Time Distributed Linear Equation Solver for Minimum Norm Solutions
This paper proposes distributed algorithms for multi-agent networks to
achieve a solution in finite time to a linear equation where has
full row rank, and with the minimum -norm in the underdetermined case
(where 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 , which is the minimum -norm solution in the
underdetermined case. Numerical simulations are also provided as validation of
the proposed algorithms
A Distributed Algorithm for Solving Linear Algebraic Equations Over Random Networks
In this paper, we consider the problem of solving linear algebraic equations
of the form among multi agents which seek a solution by using local
information in presence of random communication topologies. The equation is
solved by agents where each agent only knows a subset of rows of the
partitioned matrix . We formulate the problem such that this formulation
does not need the distribution of random interconnection graphs. Therefore,
this framework includes asynchronous updates or unreliable communication
protocols without B-connectivity assumption. We apply the random
Krasnoselskii-Mann iterative algorithm which converges almost surely and in
mean square to a solution of the problem for any matrices and and any
initial conditions of agents' states. We demonestrate that the limit point to
which the agents' states converge is determined by the unique solution of a
convex optimization problem regardless of the distribution of random
communication graphs. Eventually, we show by two numerical examples that the
rate of convergence of the algorithm cannot be guaranteed.Comment: 10 pages, 2 figures, a preliminary version of this paper appears
without proofs in the Proceedings of the 57th IEEE Conference on Decision and
Control, Miami Beach, FL, USA, December 17-19, 201
Network Flows that Solve Linear Equations
We study distributed network flows as solvers in continuous time for the
linear algebraic equation . Each node has
access to a row of the matrix and the
corresponding entry in the vector . The first "consensus +
projection" flow under investigation consists of two terms, one from standard
consensus dynamics and the other contributing to projection onto each affine
subspace specified by the and . The second "projection
consensus" flow on the other hand simply replaces the relative state feedback
in consensus dynamics with projected relative state feedback. Without
dwell-time assumption on switching graphs as well as without positively lower
bounded assumption on arc weights, we prove that all node states converge to a
common solution of the linear algebraic equation, if there is any. The
convergence is global for the "consensus + projection" flow while local for the
"projection consensus" flow in the sense that the initial values must lie on
the affine subspaces. If the linear equation has no exact solutions, we show
that the node states can converge to a ball around the least squares solution
whose radius can be made arbitrarily small through selecting a sufficiently
large gain for the "consensus + projection" flow under fixed bidirectional
graphs. Semi-global convergence to approximate least squares solutions is
demonstrated for general switching directed graphs under suitable conditions.
It is also shown that the "projection consensus" flow drives the average of the
node states to the least squares solution with complete graph. Numerical
examples are provided as illustrations of the established results.Comment: IEEE Transactions on Automatic Control, in pres
Distributed Computation of Linear Matrix Equations: An Optimization Perspective
This paper investigates the distributed computation of the well-known linear
matrix equation in the form of , with the matrices A, B, X, and F of
appropriate dimensions, over multi-agent networks from an optimization
perspective. In this paper, we consider the standard distributed
matrix-information structures, where each agent of the considered multi-agent
network has access to one of the sub-block matrices of A, B, and F. To be
specific, we first propose different decomposition methods to reformulate the
matrix equations in standard structures as distributed constrained optimization
problems by introducing substitutional variables; we show that the solutions of
the reformulated distributed optimization problems are equivalent to least
squares solutions to original matrix equations; and we design distributed
continuous-time algorithms for the constrained optimization problems, even by
using augmented matrices and a derivative feedback technique. Moreover, we
prove the exponential convergence of the algorithms to a least squares solution
to the matrix equation for any initial condition.Comment: This paper has been submitted to TA