4,144 research outputs found
A Sums-of-Squares Extension of Policy Iterations
In order to address the imprecision often introduced by widening operators in
static analysis, policy iteration based on min-computations amounts to
considering the characterization of reachable value set of a program as an
iterative computation of policies, starting from a post-fixpoint. Computing
each policy and the associated invariant relies on a sequence of numerical
optimizations. While the early research efforts relied on linear programming
(LP) to address linear properties of linear programs, the current state of the
art is still limited to the analysis of linear programs with at most quadratic
invariants, relying on semidefinite programming (SDP) solvers to compute
policies, and LP solvers to refine invariants.
We propose here to extend the class of programs considered through the use of
Sums-of-Squares (SOS) based optimization. Our approach enables the precise
analysis of switched systems with polynomial updates and guards. The analysis
presented has been implemented in Matlab and applied on existing programs
coming from the system control literature, improving both the range of
analyzable systems and the precision of previously handled ones.Comment: 29 pages, 4 figure
Linear Hamilton Jacobi Bellman Equations in High Dimensions
The Hamilton Jacobi Bellman Equation (HJB) provides the globally optimal
solution to large classes of control problems. Unfortunately, this generality
comes at a price, the calculation of such solutions is typically intractible
for systems with more than moderate state space size due to the curse of
dimensionality. This work combines recent results in the structure of the HJB,
and its reduction to a linear Partial Differential Equation (PDE), with methods
based on low rank tensor representations, known as a separated representations,
to address the curse of dimensionality. The result is an algorithm to solve
optimal control problems which scales linearly with the number of states in a
system, and is applicable to systems that are nonlinear with stochastic forcing
in finite-horizon, average cost, and first-exit settings. The method is
demonstrated on inverted pendulum, VTOL aircraft, and quadcopter models, with
system dimension two, six, and twelve respectively.Comment: 8 pages. Accepted to CDC 201
Successive normalization of rectangular arrays
Standard statistical techniques often require transforming data to have mean
and standard deviation . Typically, this process of "standardization" or
"normalization" is applied across subjects when each subject produces a single
number. High throughput genomic and financial data often come as rectangular
arrays where each coordinate in one direction concerns subjects who might have
different status (case or control, say), and each coordinate in the other
designates "outcome" for a specific feature, for example, "gene," "polymorphic
site" or some aspect of financial profile. It may happen, when analyzing data
that arrive as a rectangular array, that one requires BOTH the subjects and the
features to be "on the same footing." Thus there may be a need to standardize
across rows and columns of the rectangular matrix. There arises the question as
to how to achieve this double normalization. We propose and investigate the
convergence of what seems to us a natural approach to successive normalization
which we learned from our colleague Bradley Efron. We also study the
implementation of the method on simulated data and also on data that arose from
scientific experimentation.Comment: Published in at http://dx.doi.org/10.1214/09-AOS743 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org). With Correction
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