1,984 research outputs found
On Difference-of-SOS and Difference-of-Convex-SOS Decompositions for Polynomials
In this paper, we are interested in developing polynomial decomposition
techniques to reformulate real valued multivariate polynomials into
difference-of-sums-of-squares (namely, D-SOS) and
difference-of-convex-sums-of-squares (namely, DC-SOS). Firstly, we prove that
the set of D-SOS and DC-SOS polynomials are vector spaces and equivalent to the
set of real valued polynomials. Moreover, the problem of finding D-SOS and
DC-SOS decompositions are equivalent to semidefinite programs (SDP) which can
be solved to any desired precision in polynomial time. Some important algebraic
properties and the relationships among the set of sums-of-squares (SOS)
polynomials, positive semidefinite (PSD) polynomials, convex-sums-of-squares
(CSOS) polynomials, SOS-convex polynomials, D-SOS and DC-SOS polynomials are
discussed. Secondly, we focus on establishing several practical algorithms for
constructing D-SOS and DC-SOS decompositions for any polynomial without solving
SDP. Using DC-SOS decomposition, we can reformulate polynomial optimization
problems in the realm of difference-of-convex (DC) programming, which can be
handled by efficient DC programming approaches. Some examples illustrate how to
use our methods for constructing D-SOS and DC-SOS decompositions. Numerical
performance of D-SOS and DC-SOS decomposition algorithms and their parallelized
methods are tested on a synthetic dataset with 1750 randomly generated large
and small sized sparse and dense polynomials. Some real-world applications in
higher order moment portfolio optimization problems, eigenvalue complementarity
problems, Euclidean distance matrix completion problems, and Boolean polynomial
programs are also presented.Comment: 47 pages, 19 figure
Coordinate shadows of semi-definite and Euclidean distance matrices
We consider the projected semi-definite and Euclidean distance cones onto a
subset of the matrix entries. These two sets are precisely the input data
defining feasible semi-definite and Euclidean distance completion problems. We
classify when these sets are closed, and use the boundary structure of these
two sets to elucidate the Krislock-Wolkowicz facial reduction algorithm. In
particular, we show that under a chordality assumption, the "minimal cones" of
these problems admit combinatorial characterizations. As a byproduct, we record
a striking relationship between the complexity of the general facial reduction
algorithm (singularity degree) and facial exposedness of conic images under a
linear mapping.Comment: 21 page
Euclidean Distance Matrices: Essential Theory, Algorithms and Applications
Euclidean distance matrices (EDM) are matrices of squared distances between
points. The definition is deceivingly simple: thanks to their many useful
properties they have found applications in psychometrics, crystallography,
machine learning, wireless sensor networks, acoustics, and more. Despite the
usefulness of EDMs, they seem to be insufficiently known in the signal
processing community. Our goal is to rectify this mishap in a concise tutorial.
We review the fundamental properties of EDMs, such as rank or
(non)definiteness. We show how various EDM properties can be used to design
algorithms for completing and denoising distance data. Along the way, we
demonstrate applications to microphone position calibration, ultrasound
tomography, room reconstruction from echoes and phase retrieval. By spelling
out the essential algorithms, we hope to fast-track the readers in applying
EDMs to their own problems. Matlab code for all the described algorithms, and
to generate the figures in the paper, is available online. Finally, we suggest
directions for further research.Comment: - 17 pages, 12 figures, to appear in IEEE Signal Processing Magazine
- change of title in the last revisio
Euclidean distance geometry and applications
Euclidean distance geometry is the study of Euclidean geometry based on the
concept of distance. This is useful in several applications where the input
data consists of an incomplete set of distances, and the output is a set of
points in Euclidean space that realizes the given distances. We survey some of
the theory of Euclidean distance geometry and some of the most important
applications: molecular conformation, localization of sensor networks and
statics.Comment: 64 pages, 21 figure
- …