1,350 research outputs found
A distributionally robust perspective on uncertainty quantification and chance constrained programming
The objective of uncertainty quantification is to certify that a given physical, engineering or economic system satisfies multiple safety conditions with high probability. A more ambitious goal is to actively influence the system so as to guarantee and maintain its safety, a scenario which can be modeled through a chance constrained program. In this paper we assume that the parameters of the system are governed by an ambiguous distribution that is only known to belong to an ambiguity set characterized through generalized moment bounds and structural properties such as symmetry, unimodality or independence patterns. We delineate the watershed between tractability and intractability in ambiguity-averse uncertainty quantification and chance constrained programming. Using tools from distributionally robust optimization, we derive explicit conic reformulations for tractable problem classes and suggest efficiently computable conservative approximations for intractable ones
A complete family of separability criteria
We introduce a new family of separability criteria that are based on the
existence of extensions of a bipartite quantum state to a larger number
of parties satisfying certain symmetry properties. It can be easily shown that
all separable states have the required extensions, so the non-existence of such
an extension for a particular state implies that the state is entangled. One of
the main advantages of this approach is that searching for the extension can be
cast as a convex optimization problem known as a semidefinite program (SDP).
Whenever an extension does not exist, the dual optimization constructs an
explicit entanglement witness for the particular state. These separability
tests can be ordered in a hierarchical structure whose first step corresponds
to the well-known Positive Partial Transpose (Peres-Horodecki) criterion, and
each test in the hierarchy is at least as powerful as the preceding one. This
hierarchy is complete, in the sense that any entangled state is guaranteed to
fail a test at some finite point in the hierarchy, thus showing it is
entangled. The entanglement witnesses corresponding to each step of the
hierarchy have well-defined and very interesting algebraic properties that in
turn allow for a characterization of the interior of the set of positive maps.
Coupled with some recent results on the computational complexity of the
separability problem, which has been shown to be NP-hard, this hierarchy of
tests gives a complete and also computationally and theoretically appealing
characterization of mixed bipartite entangled states.Comment: 21 pages. Expanded introduction. References added, typos corrected.
Accepted for publication in Physical Review
PALP - a User Manual
This article provides a complete user's guide to version 2.1 of the toric
geometry package PALP by Maximilian Kreuzer and others. In particular,
previously undocumented applications such as the program nef.x are discussed in
detail. New features of PALP 2.1 include an extension of the program mori.x
which can now compute Mori cones and intersection rings of arbitrary dimension
and can also take specific triangulations of reflexive polytopes as input.
Furthermore, the program nef.x is enhanced by an option that allows the user to
enter reflexive Gorenstein cones as input. The present documentation is
complemented by a Wiki which is available online.Comment: 71 pages, to appear in "Strings, Gauge Fields, and the Geometry
Behind - The Legacy of Maximilian Kreuzer". PALP Wiki available at
http://palp.itp.tuwien.ac.at/wiki/index.php/Main_Pag
Finding a Spherically Symmetric Cosmology from Observations in Observational Coordinates -- Advantages and Challenges
One of the continuing challenges in cosmology has been to determine the
large-scale space-time metric from observations with a minimum of assumptions
-- without, for instance, assuming that the universe is almost
Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW). If we are lucky enough this
would be a way of demonstrating that our universe is FLRW, instead of
presupposing it or simply showing that the observations are consistent with
FLRW. Showing how to do this within the more general spherically symmetric,
inhomogeneous space-time framework takes us a long way towards fulfilling this
goal. In recent work researchers have shown how this can be done both in the
traditional Lema\^{i}tre-Tolman-Bondi (LTB) 3 + 1 coordinate framework, and in
the observational coordinate (OC) framework. In this paper we investigate the
stability of solutions, and the use of data in the OC field equations including
their time evolution and compare both approaches with respect to the
singularity problem at the maximum of the angular-diameter distance, the
stability of solutions, and the use of data in the field equations. This allows
a more detailed account and assessment of the OC integration procedure, and
enables a comparison of the relative advantages of the two equivalent solution
frameworks. Both formulations and integration procedures should, in principle,
lead to the same results. However, as we show in this paper, the OC procedure
manifests certain advantages, particularly in the avoidance of coordinate
singularities at the maximum of the angular-diameter distance, and in the
stability of the solutions obtained. This particular feature is what allows us
to do the best fitting of the data to smooth data functions and the possibility
of constructing analytic solutions to the field equations.Comment: 31 page
Plethysm and lattice point counting
We apply lattice point counting methods to compute the multiplicities in the
plethysm of . Our approach gives insight into the asymptotic growth of
the plethysm and makes the problem amenable to computer algebra. We prove an
old conjecture of Howe on the leading term of plethysm. For any partition
of 3,4, or 5 we obtain an explicit formula in and for the
multiplicity of in .Comment: 25 pages including appendix, 1 figure, computational results and code
available at http://thomas-kahle.de/plethysm.html, v2: various improvements,
v3: final version appeared in JFoC
Real Algebraic Geometry with a View Toward Hyperbolic Programming and Free Probability
Continuing the tradition initiated in the MFO workshops held in 2014 and 2017, this workshop was dedicated to the newest developments in real algebraic geometry and polynomial optimization, with a particular emphasis on free non-commutative real algebraic geometry and hyperbolic programming. A particular effort was invested in exploring the interrelations with free probability. This established an interesting dialogue between researchers working in real algebraic geometry and those working in free probability, from which emerged new exciting and promising synergies
A Class of Semidefinite Programs with rank-one solutions
We show that a class of semidefinite programs (SDP) admits a solution that is
a positive semidefinite matrix of rank at most , where is the rank of
the matrix involved in the objective function of the SDP. The optimization
problems of this class are semidefinite packing problems, which are the SDP
analogs to vector packing problems. Of particular interest is the case in which
our result guarantees the existence of a solution of rank one: we show that the
computation of this solution actually reduces to a Second Order Cone Program
(SOCP). We point out an application in statistics, in the optimal design of
experiments.Comment: 16 page
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