3,629 research outputs found
Optimal Analysis of Discrete-time Affine Systems
Our very first concern is the resolution of the verification problem for the
class of discrete-time affine dynamical systems. This verification problem is
turned into an optimization problem where the constraint set is the reachable
values set of the dynamical system. To solve this optimization problem, we
truncate the infinite sequences belonging to the reachable values set at some
step which is uniform with respect to the initial conditions. In theory, the
best possible uniform step is the optimal solution of a non-convex
semi-definite program. In practice, we propose a methodology to compute a
uniform step that over-approximate the best solution.Comment: 16 page
Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving
We derive efficient algorithms for coarse approximation of algebraic
hypersurfaces, useful for estimating the distance between an input polynomial
zero set and a given query point. Our methods work best on sparse polynomials
of high degree (in any number of variables) but are nevertheless completely
general. The underlying ideas, which we take the time to describe in an
elementary way, come from tropical geometry. We thus reduce a hard algebraic
problem to high-precision linear optimization, proving new upper and lower
complexity estimates along the way.Comment: 15 pages, 9 figures. Submitted to a conference proceeding
Solving rank-constrained semidefinite programs in exact arithmetic
We consider the problem of minimizing a linear function over an affine
section of the cone of positive semidefinite matrices, with the additional
constraint that the feasible matrix has prescribed rank. When the rank
constraint is active, this is a non-convex optimization problem, otherwise it
is a semidefinite program. Both find numerous applications especially in
systems control theory and combinatorial optimization, but even in more general
contexts such as polynomial optimization or real algebra. While numerical
algorithms exist for solving this problem, such as interior-point or
Newton-like algorithms, in this paper we propose an approach based on symbolic
computation. We design an exact algorithm for solving rank-constrained
semidefinite programs, whose complexity is essentially quadratic on natural
degree bounds associated to the given optimization problem: for subfamilies of
the problem where the size of the feasible matrix is fixed, the complexity is
polynomial in the number of variables. The algorithm works under assumptions on
the input data: we prove that these assumptions are generically satisfied. We
also implement it in Maple and discuss practical experiments.Comment: Published at ISSAC 2016. Extended version submitted to the Journal of
Symbolic Computatio
Autocalibration with the Minimum Number of Cameras with Known Pixel Shape
In 3D reconstruction, the recovery of the calibration parameters of the
cameras is paramount since it provides metric information about the observed
scene, e.g., measures of angles and ratios of distances. Autocalibration
enables the estimation of the camera parameters without using a calibration
device, but by enforcing simple constraints on the camera parameters. In the
absence of information about the internal camera parameters such as the focal
length and the principal point, the knowledge of the camera pixel shape is
usually the only available constraint. Given a projective reconstruction of a
rigid scene, we address the problem of the autocalibration of a minimal set of
cameras with known pixel shape and otherwise arbitrarily varying intrinsic and
extrinsic parameters. We propose an algorithm that only requires 5 cameras (the
theoretical minimum), thus halving the number of cameras required by previous
algorithms based on the same constraint. To this purpose, we introduce as our
basic geometric tool the six-line conic variety (SLCV), consisting in the set
of planes intersecting six given lines of 3D space in points of a conic. We
show that the set of solutions of the Euclidean upgrading problem for three
cameras with known pixel shape can be parameterized in a computationally
efficient way. This parameterization is then used to solve autocalibration from
five or more cameras, reducing the three-dimensional search space to a
two-dimensional one. We provide experiments with real images showing the good
performance of the technique.Comment: 19 pages, 14 figures, 7 tables, J. Math. Imaging Vi
Robust stability at the Swallowtail singularity
Consider the set of monic fourth-order real polynomials transformed so that the constant term is one. In the three-dimensional space of the coefficients describing this set, the domain of asymptotic stability is bounded by a surface with the Whitney umbrella singularity. The maximum of the real parts of the roots of these polynomials is globally minimized at the Swallowtail singular point of the discriminant surface of the set corresponding to a negative real root of multiplicity four. Motivated by this example, we review recent works on robust stability, abscissa optimization, heavily damped systems, dissipation-induced instabilities, and eigenvalue dynamics in order to point out some connections that appear to be not widely known
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