115 research outputs found
Sparse sum-of-squares certificates on finite abelian groups
Let G be a finite abelian group. This paper is concerned with nonnegative
functions on G that are sparse with respect to the Fourier basis. We establish
combinatorial conditions on subsets S and T of Fourier basis elements under
which nonnegative functions with Fourier support S are sums of squares of
functions with Fourier support T. Our combinatorial condition involves
constructing a chordal cover of a graph related to G and S (the Cayley graph
Cay(,S)) with maximal cliques related to T. Our result relies on two
main ingredients: the decomposition of sparse positive semidefinite matrices
with a chordal sparsity pattern, as well as a simple but key observation
exploiting the structure of the Fourier basis elements of G.
We apply our general result to two examples. First, in the case where , by constructing a particular chordal cover of the half-cube
graph, we prove that any nonnegative quadratic form in n binary variables is a
sum of squares of functions of degree at most , establishing
a conjecture of Laurent. Second, we consider nonnegative functions of degree d
on (when d divides N). By constructing a particular chordal
cover of the d'th power of the N-cycle, we prove that any such function is a
sum of squares of functions with at most nonzero Fourier
coefficients. Dually this shows that a certain cyclic polytope in
with N vertices can be expressed as a projection of a section
of the cone of psd matrices of size . Putting gives a
family of polytopes with LP extension complexity
and SDP extension complexity
. To the best of our knowledge, this is the
first explicit family of polytopes in increasing dimensions where
.Comment: 34 page
Fourier sum of squares certificates
The non-negativity of a function on a finite abelian group can be certified
by its Fourier sum of squares (FSOS). In this paper, we propose a method of
certifying the non-negativity of an integer-valued function by an FSOS
certificate, which is defined to be an FSOS with a small error. We prove the
existence of exponentially sparse polynomial and rational FSOS certificates and
we provide two methods to validate them. As a consequence of the aforementioned
existence theorems, we propose a semidefinite programming (SDP)-based algorithm
to efficiently compute a sparse FSOS certificate. For applications, we consider
certificate problems for maximum satisfiability (MAX-SAT) and maximum
k-colorable subgraph (MkCS) and demonstrate our theoretical results and
algorithm by numerical experiments
Equivariant semidefinite lifts and sum-of-squares hierarchies
A central question in optimization is to maximize (or minimize) a linear
function over a given polytope P. To solve such a problem in practice one needs
a concise description of the polytope P. In this paper we are interested in
representations of P using the positive semidefinite cone: a positive
semidefinite lift (psd lift) of a polytope P is a representation of P as the
projection of an affine slice of the positive semidefinite cone
. Such a representation allows linear optimization problems
over P to be written as semidefinite programs of size d. Such representations
can be beneficial in practice when d is much smaller than the number of facets
of the polytope P. In this paper we are concerned with so-called equivariant
psd lifts (also known as symmetric psd lifts) which respect the symmetries of
the polytope P. We present a representation-theoretic framework to study
equivariant psd lifts of a certain class of symmetric polytopes known as
orbitopes. Our main result is a structure theorem where we show that any
equivariant psd lift of size d of an orbitope is of sum-of-squares type where
the functions in the sum-of-squares decomposition come from an invariant
subspace of dimension smaller than d^3. We use this framework to study two
well-known families of polytopes, namely the parity polytope and the cut
polytope, and we prove exponential lower bounds for equivariant psd lifts of
these polytopes.Comment: v2: 30 pages, Minor changes in presentation; v3: 29 pages, New
structure theorem for general orbitopes + changes in presentatio
Finding Significant Fourier Coefficients: Clarifications, Simplifications, Applications and Limitations
Ideas from Fourier analysis have been used in cryptography for the last three
decades. Akavia, Goldwasser and Safra unified some of these ideas to give a
complete algorithm that finds significant Fourier coefficients of functions on
any finite abelian group. Their algorithm stimulated a lot of interest in the
cryptography community, especially in the context of `bit security'. This
manuscript attempts to be a friendly and comprehensive guide to the tools and
results in this field. The intended readership is cryptographers who have heard
about these tools and seek an understanding of their mechanics and their
usefulness and limitations. A compact overview of the algorithm is presented
with emphasis on the ideas behind it. We show how these ideas can be extended
to a `modulus-switching' variant of the algorithm. We survey some applications
of this algorithm, and explain that several results should be taken in the
right context. In particular, we point out that some of the most important bit
security problems are still open. Our original contributions include: a
discussion of the limitations on the usefulness of these tools; an answer to an
open question about the modular inversion hidden number problem
Recommended from our members
Reelle Algebraische Geometrie
This workshop was organized by Michel Coste (Rennes), Claus Scheiderer (Konstanz) and Niels Schwartz (Passau). The talks focussed on recent developments in real enumerative and tropical geometry, positivity and sums of squares, real aspects of classical algebraic geometry, semialgebraic and tame geometry, and topology and singularities of real varieties
Scalable computation of intracellular metabolite concentrations
Current mathematical frameworks for predicting the flux state and
macromolecular composition of the cell do not rely on thermodynamic constraints
to determine the spontaneous direction of reactions. These predictions may be
biologically infeasible as a result. Imposing thermodynamic constraints
requires accurate estimations of intracellular metabolite concentrations. These
concentrations are constrained within physiologically possible ranges to enable
an organism to grow in extreme conditions and adapt to its environment. Here,
we introduce tractable computational techniques to characterize intracellular
metabolite concentrations within a constraint-based modeling framework. This
model provides a feasible concentration set, which can generally be nonconvex
and disconnected. We examine three approaches based on polynomial optimization,
random sampling, and global optimization. We leverage the sparsity and
algebraic structure of the underlying biophysical models to enhance the
computational efficiency of these techniques. We then compare their performance
in two case studies, showing that the global-optimization formulation exhibits
more desirable scaling properties than the random-sampling and
polynomial-optimization formulation, and, thus, is a promising candidate for
handling large-scale metabolic networks
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