1,819 research outputs found
Information Acquisition with Sensing Robots: Algorithms and Error Bounds
Utilizing the capabilities of configurable sensing systems requires
addressing difficult information gathering problems. Near-optimal approaches
exist for sensing systems without internal states. However, when it comes to
optimizing the trajectories of mobile sensors the solutions are often greedy
and rarely provide performance guarantees. Notably, under linear Gaussian
assumptions, the problem becomes deterministic and can be solved off-line.
Approaches based on submodularity have been applied by ignoring the sensor
dynamics and greedily selecting informative locations in the environment. This
paper presents a non-greedy algorithm with suboptimality guarantees, which does
not rely on submodularity and takes the sensor dynamics into account. Our
method performs provably better than the widely used greedy one. Coupled with
linearization and model predictive control, it can be used to generate adaptive
policies for mobile sensors with non-linear sensing models. Applications in gas
concentration mapping and target tracking are presented.Comment: 9 pages (two-column); 2 figures; Manuscript submitted to the 2014
IEEE International Conference on Robotics and Automatio
On linear convergence of a distributed dual gradient algorithm for linearly constrained separable convex problems
In this paper we propose a distributed dual gradient algorithm for minimizing
linearly constrained separable convex problems and analyze its rate of
convergence. In particular, we prove that under the assumption of strong
convexity and Lipshitz continuity of the gradient of the primal objective
function we have a global error bound type property for the dual problem. Using
this error bound property we devise a fully distributed dual gradient scheme,
i.e. a gradient scheme based on a weighted step size, for which we derive
global linear rate of convergence for both dual and primal suboptimality and
for primal feasibility violation. Many real applications, e.g. distributed
model predictive control, network utility maximization or optimal power flow,
can be posed as linearly constrained separable convex problems for which dual
gradient type methods from literature have sublinear convergence rate. In the
present paper we prove for the first time that in fact we can achieve linear
convergence rate for such algorithms when they are used for solving these
applications. Numerical simulations are also provided to confirm our theory.Comment: 14 pages, 4 figures, submitted to Automatica Journal, February 2014.
arXiv admin note: substantial text overlap with arXiv:1401.4398. We revised
the paper, adding more simulations and checking for typo
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Least quantile regression via modern optimization
We address the Least Quantile of Squares (LQS) (and in particular the Least
Median of Squares) regression problem using modern optimization methods. We
propose a Mixed Integer Optimization (MIO) formulation of the LQS problem which
allows us to find a provably global optimal solution for the LQS problem. Our
MIO framework has the appealing characteristic that if we terminate the
algorithm early, we obtain a solution with a guarantee on its sub-optimality.
We also propose continuous optimization methods based on first-order
subdifferential methods, sequential linear optimization and hybrid combinations
of them to obtain near optimal solutions to the LQS problem. The MIO algorithm
is found to benefit significantly from high quality solutions delivered by our
continuous optimization based methods. We further show that the MIO approach
leads to (a) an optimal solution for any dataset, where the data-points
's are not necessarily in general position, (b) a simple
proof of the breakdown point of the LQS objective value that holds for any
dataset and (c) an extension to situations where there are polyhedral
constraints on the regression coefficient vector. We report computational
results with both synthetic and real-world datasets showing that the MIO
algorithm with warm starts from the continuous optimization methods solve small
() and medium () size problems to provable optimality in under
two hours, and outperform all publicly available methods for large-scale
(10,000) LQS problems.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1223 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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