36,217 research outputs found
Randomized Optimal Consensus of Multi-agent Systems
In this paper, we formulate and solve a randomized optimal consensus problem
for multi-agent systems with stochastically time-varying interconnection
topology. The considered multi-agent system with a simple randomized iterating
rule achieves an almost sure consensus meanwhile solving the optimization
problem \min_{z\in \mathds{R}^d}\ \sum_{i=1}^n f_i(z), in which the optimal
solution set of objective function can only be observed by agent
itself. At each time step, simply determined by a Bernoulli trial, each agent
independently and randomly chooses either taking an average among its neighbor
set, or projecting onto the optimal solution set of its own optimization
component. Both directed and bidirectional communication graphs are studied.
Connectivity conditions are proposed to guarantee an optimal consensus almost
surely with proper convexity and intersection assumptions. The convergence
analysis is carried out using convex analysis. We compare the randomized
algorithm with the deterministic one via a numerical example. The results
illustrate that a group of autonomous agents can reach an optimal opinion by
each node simply making a randomized trade-off between following its neighbors
or sticking to its own opinion at each time step
Randomized Constraints Consensus for Distributed Robust Linear Programming
In this paper we consider a network of processors aiming at cooperatively
solving linear programming problems subject to uncertainty. Each node only
knows a common cost function and its local uncertain constraint set. We propose
a randomized, distributed algorithm working under time-varying, asynchronous
and directed communication topology. The algorithm is based on a local
computation and communication paradigm. At each communication round, nodes
perform two updates: (i) a verification in which they check-in a randomized
setup-the robust feasibility (and hence optimality) of the candidate optimal
point, and (ii) an optimization step in which they exchange their candidate
bases (minimal sets of active constraints) with neighbors and locally solve an
optimization problem whose constraint set includes: a sampled constraint
violating the candidate optimal point (if it exists), agent's current basis and
the collection of neighbor's basis. As main result, we show that if a processor
successfully performs the verification step for a sufficient number of
communication rounds, it can stop the algorithm since a consensus has been
reached. The common solution is-with high confidence-feasible (and hence
optimal) for the entire set of uncertainty except a subset having arbitrary
small probability measure. We show the effectiveness of the proposed
distributed algorithm on a multi-core platform in which the nodes communicate
asynchronously.Comment: Accepted for publication in the 20th World Congress of the
International Federation of Automatic Control (IFAC
Distributed Optimization: Convergence Conditions from a Dynamical System Perspective
This paper explores the fundamental properties of distributed minimization of
a sum of functions with each function only known to one node, and a
pre-specified level of node knowledge and computational capacity. We define the
optimization information each node receives from its objective function, the
neighboring information each node receives from its neighbors, and the
computational capacity each node can take advantage of in controlling its
state. It is proven that there exist a neighboring information way and a
control law that guarantee global optimal consensus if and only if the solution
sets of the local objective functions admit a nonempty intersection set for
fixed strongly connected graphs. Then we show that for any tolerated error, we
can find a control law that guarantees global optimal consensus within this
error for fixed, bidirectional, and connected graphs under mild conditions. For
time-varying graphs, we show that optimal consensus can always be achieved as
long as the graph is uniformly jointly strongly connected and the nonempty
intersection condition holds. The results illustrate that nonempty intersection
for the local optimal solution sets is a critical condition for successful
distributed optimization for a large class of algorithms
Distributed Partitioned Big-Data Optimization via Asynchronous Dual Decomposition
In this paper we consider a novel partitioned framework for distributed
optimization in peer-to-peer networks. In several important applications the
agents of a network have to solve an optimization problem with two key
features: (i) the dimension of the decision variable depends on the network
size, and (ii) cost function and constraints have a sparsity structure related
to the communication graph. For this class of problems a straightforward
application of existing consensus methods would show two inefficiencies: poor
scalability and redundancy of shared information. We propose an asynchronous
distributed algorithm, based on dual decomposition and coordinate methods, to
solve partitioned optimization problems. We show that, by exploiting the
problem structure, the solution can be partitioned among the nodes, so that
each node just stores a local copy of a portion of the decision variable
(rather than a copy of the entire decision vector) and solves a small-scale
local problem
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