14,334 research outputs found
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
A Finite-Time Cutting Plane Algorithm for Distributed Mixed Integer Linear Programming
Many problems of interest for cyber-physical network systems can be
formulated as Mixed Integer Linear Programs in which the constraints are
distributed among the agents. In this paper we propose a distributed algorithm
to solve this class of optimization problems in a peer-to-peer network with no
coordinator and with limited computation and communication capabilities. In the
proposed algorithm, at each communication round, agents solve locally a small
LP, generate suitable cutting planes, namely intersection cuts and cost-based
cuts, and communicate a fixed number of active constraints, i.e., a candidate
optimal basis. We prove that, if the cost is integer, the algorithm converges
to the lexicographically minimal optimal solution in a finite number of
communication rounds. Finally, through numerical computations, we analyze the
algorithm convergence as a function of the network size.Comment: 6 pages, 3 figure
Stable Camera Motion Estimation Using Convex Programming
We study the inverse problem of estimating n locations (up to
global scale, translation and negation) in from noisy measurements of a
subset of the (unsigned) pairwise lines that connect them, that is, from noisy
measurements of for some pairs (i,j) (where the
signs are unknown). This problem is at the core of the structure from motion
(SfM) problem in computer vision, where the 's represent camera locations
in . The noiseless version of the problem, with exact line measurements,
has been considered previously under the general title of parallel rigidity
theory, mainly in order to characterize the conditions for unique realization
of locations. For noisy pairwise line measurements, current methods tend to
produce spurious solutions that are clustered around a few locations. This
sensitivity of the location estimates is a well-known problem in SfM,
especially for large, irregular collections of images.
In this paper we introduce a semidefinite programming (SDP) formulation,
specially tailored to overcome the clustering phenomenon. We further identify
the implications of parallel rigidity theory for the location estimation
problem to be well-posed, and prove exact (in the noiseless case) and stable
location recovery results. We also formulate an alternating direction method to
solve the resulting semidefinite program, and provide a distributed version of
our formulation for large numbers of locations. Specifically for the camera
location estimation problem, we formulate a pairwise line estimation method
based on robust camera orientation and subspace estimation. Lastly, we
demonstrate the utility of our algorithm through experiments on real images.Comment: 40 pages, 12 figures, 6 tables; notation and some unclear parts
updated, some typos correcte
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