15,009 research outputs found
Global optimization of polynomials using gradient tentacles and sums of squares
In this work, the combine the theory of generalized critical values with the
theory of iterated rings of bounded elements (real holomorphy rings).
We consider the problem of computing the global infimum of a real polynomial
in several variables. Every global minimizer lies on the gradient variety. If
the polynomial attains a minimum, it is therefore equivalent to look for the
greatest lower bound on its gradient variety. Nie, Demmel and Sturmfels proved
recently a theorem about the existence of sums of squares certificates for such
lower bounds. Based on these certificates, they find arbitrarily tight
relaxations of the original problem that can be formulated as semidefinite
programs and thus be solved efficiently.
We deal here with the more general case when the polynomial is bounded from
belo w but does not necessarily attain a minimum. In this case, the method of
Nie, Demmel and Sturmfels might yield completely wrong results. In order to
overcome this problem, we replace the gradient variety by larger semialgebraic
sets which we call gradient tentacles. It now gets substantially harder to
prove the existence of the necessary sums of squares certificates.Comment: 22 page
The Moment Problem for Continuous Positive Semidefinite Linear functionals
Let be a locally convex topology on the countable dimensional
polynomial -algebra \rx:=\reals[X_1,...,X_n]. Let be a closed
subset of , and let be a finitely generated
quadratic module in \rx. We investigate the following question: When is the
cone \Pos(K) (of polynomials nonnegative on ) included in the closure of
? We give an interpretation of this inclusion with respect to representing
continuous linear functionals by measures. We discuss several examples; we
compute the closure of M=\sos with respect to weighted norm- topologies.
We show that this closure coincides with the cone \Pos(K) where is a
certain convex compact polyhedron.Comment: 14 page
Maximizing the number of nonnegative subsets
Given a set of real numbers, if the sum of elements of every subset of
size larger than is negative, what is the maximum number of subsets of
nonnegative sum? In this note we show that the answer is , settling a problem of Tsukerman.
We provide two proofs, the first establishes and applies a weighted version of
Hall's Theorem and the second is based on an extension of the nonuniform
Erd\H{o}s-Ko-Rado Theorem
Nonnegative polynomials and their Carath\'eodory number
In 1888 Hilbert showed that every nonnegative homogeneous polynomial with
real coefficients of degree in variables is a sum of squares if and
only if (quadratic forms), (binary forms) or (ternary
quartics). In these cases, it is interesting to compute canonical expressions
for these decompositions. Starting from Carath\'eodory's Theorem, we compute
the Carath\'eodory number of Hilbert cones of nonnegative quadratic and binary
forms.Comment: 9 pages. Discrete & Computational Geometry (2014
Nonnegative k-sums, fractional covers, and probability of small deviations
More than twenty years ago, Manickam, Mikl\'{o}s, and Singhi conjectured that
for any integers satisfying , every set of real numbers
with nonnegative sum has at least -element subsets whose
sum is also nonnegative. In this paper we discuss the connection of this
problem with matchings and fractional covers of hypergraphs, and with the
question of estimating the probability that the sum of nonnegative independent
random variables exceeds its expectation by a given amount. Using these
connections together with some probabilistic techniques, we verify the
conjecture for . This substantially improves the best previously
known exponential lower bound . In addition we prove
a tight stability result showing that for every and all sufficiently large
, every set of reals with a nonnegative sum that does not contain a
member whose sum with any other members is nonnegative, contains at least
subsets of cardinality with
nonnegative sum.Comment: 15 pages, a section of Hilton-Milner type result adde
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