2,815 research outputs found
Real radical initial ideals
We explore the consequences of an ideal I of real polynomials having a real
radical initial ideal, both for the geometry of the real variety of I and as an
application to sums of squares representations of polynomials. We show that if
in_w(I) is real radical for a vector w in the tropical variety, then w is in
the logarithmic set of the real variety. We also give algebraic sufficient
conditions for w to be in the logarithmic limit set of a more general
semialgebraic set. If in addition the entries of w are positive, then the
corresponding quadratic module is stable. In particular, if in_w(I) is real
radical for some positive vector w then the set of sums of squares modulo I is
stable. This provides a method for checking the conditions for stability given
by Powers and Scheiderer.Comment: 16 pages, added examples, minor revision
Toric completions and bounded functions on real algebraic varieties
Given a semi-algebraic set S, we study compactifications of S that arise from
embeddings into complete toric varieties. This makes it possible to describe
the asymptotic growth of polynomial functions on S in terms of combinatorial
data. We extend our earlier work to compute the ring of bounded functions in
this setting and discuss applications to positive polynomials and the moment
problem. Complete results are obtained in special cases, like sets defined by
binomial inequalities. We also show that the wild behaviour of certain examples
constructed by Krug and by Mondal-Netzer cannot occur in a toric setting.Comment: 19 pages; minor updates and correction
Geometry of physical dispersion relations
To serve as a dispersion relation, a cotangent bundle function must satisfy
three simple algebraic properties. These conditions are derived from the
inescapable physical requirements to have predictive matter field dynamics and
an observer-independent notion of positive energy. Possible modifications of
the standard relativistic dispersion relation are thereby severely restricted.
For instance, the dispersion relations associated with popular deformations of
Maxwell theory by Gambini-Pullin or Myers-Pospelov are not admissible.Comment: revised version, new section on applications added, 46 pages, 9
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Some Applications of Polynomial Optimization in Operations Research and Real-Time Decision Making
We demonstrate applications of algebraic techniques that optimize and certify
polynomial inequalities to problems of interest in the operations research and
transportation engineering communities. Three problems are considered: (i)
wireless coverage of targeted geographical regions with guaranteed signal
quality and minimum transmission power, (ii) computing real-time certificates
of collision avoidance for a simple model of an unmanned vehicle (UV)
navigating through a cluttered environment, and (iii) designing a nonlinear
hovering controller for a quadrotor UV, which has recently been used for load
transportation. On our smaller-scale applications, we apply the sum of squares
(SOS) relaxation and solve the underlying problems with semidefinite
programming. On the larger-scale or real-time applications, we use our recently
introduced "SDSOS Optimization" techniques which result in second order cone
programs. To the best of our knowledge, this is the first study of real-time
applications of sum of squares techniques in optimization and control. No
knowledge in dynamics and control is assumed from the reader
Improving Efficiency and Scalability of Sum of Squares Optimization: Recent Advances and Limitations
It is well-known that any sum of squares (SOS) program can be cast as a
semidefinite program (SDP) of a particular structure and that therein lies the
computational bottleneck for SOS programs, as the SDPs generated by this
procedure are large and costly to solve when the polynomials involved in the
SOS programs have a large number of variables and degree. In this paper, we
review SOS optimization techniques and present two new methods for improving
their computational efficiency. The first method leverages the sparsity of the
underlying SDP to obtain computational speed-ups. Further improvements can be
obtained if the coefficients of the polynomials that describe the problem have
a particular sparsity pattern, called chordal sparsity. The second method
bypasses semidefinite programming altogether and relies instead on solving a
sequence of more tractable convex programs, namely linear and second order cone
programs. This opens up the question as to how well one can approximate the
cone of SOS polynomials by second order representable cones. In the last part
of the paper, we present some recent negative results related to this question.Comment: Tutorial for CDC 201
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