18,462 research outputs found
Error bounds for monomial convexification in polynomial optimization
Convex hulls of monomials have been widely studied in the literature, and
monomial convexifications are implemented in global optimization software for
relaxing polynomials. However, there has been no study of the error in the
global optimum from such approaches. We give bounds on the worst-case error for
convexifying a monomial over subsets of . This implies additive error
bounds for relaxing a polynomial optimization problem by convexifying each
monomial separately. Our main error bounds depend primarily on the degree of
the monomial, making them easy to compute. Since monomial convexification
studies depend on the bounds on the associated variables, in the second part,
we conduct an error analysis for a multilinear monomial over two different
types of box constraints. As part of this analysis, we also derive the convex
hull of a multilinear monomial over .Comment: 33 pages, 2 figures, to appear in journa
On some extremalities in the approximate integration
Some extremalities for quadrature operators are proved for convex functions
of higher order. Such results are known in the numerical analysis, however they
are often proved under suitable differentiability assumptions. In our
considerations we do not use any other assumptions apart from higher order
convexity itself. The obtained inequalities refine the inequalities of Hadamard
type. They are applied to give error bounds of quadrature operators under the
assumptions weaker from the commonly used
Improved convergence analysis of Lasserre's measure-based upper bounds for polynomial minimization on compact sets
We consider the problem of computing the minimum value of a
polynomial over a compact set , which can be
reformulated as finding a probability measure on minimizing . Lasserre showed that it suffices to consider such measures of the form
, where is a sum-of-squares polynomial and is a given
Borel measure supported on . By bounding the degree of by one gets
a converging hierarchy of upper bounds for . When is
the hypercube , equipped with the Chebyshev measure, the parameters
are known to converge to at a rate in . We
extend this error estimate to a wider class of convex bodies, while also
allowing for a broader class of reference measures, including the Lebesgue
measure. Our analysis applies to simplices, balls and convex bodies that
locally look like a ball. In addition, we show an error estimate in when satisfies a minor geometrical condition, and in when is a convex body, equipped with the Lebesgue measure. This
improves upon the currently best known error estimates in and
for these two respective cases.Comment: 30 pages with 10 figures. Update notes for second version: Added a
new section containing numerical examples that illustrate the theoretical
results -- Fixed minor mistakes/typos -- Improved some notation -- Clarified
certain explanations in the tex
Moment-Sum-Of-Squares Approach For Fast Risk Estimation In Uncertain Environments
In this paper, we address the risk estimation problem where one aims at
estimating the probability of violation of safety constraints for a robot in
the presence of bounded uncertainties with arbitrary probability distributions.
In this problem, an unsafe set is described by level sets of polynomials that
is, in general, a non-convex set. Uncertainty arises due to the probabilistic
parameters of the unsafe set and probabilistic states of the robot. To solve
this problem, we use a moment-based representation of probability
distributions. We describe upper and lower bounds of the risk in terms of a
linear weighted sum of the moments. Weights are coefficients of a univariate
Chebyshev polynomial obtained by solving a sum-of-squares optimization problem
in the offline step. Hence, given a finite number of moments of probability
distributions, risk can be estimated in real-time. We demonstrate the
performance of the provided approach by solving probabilistic collision
checking problems where we aim to find the probability of collision of a robot
with a non-convex obstacle in the presence of probabilistic uncertainties in
the location of the robot and size, location, and geometry of the obstacle.Comment: 57th IEEE Conference on Decision and Control 201
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