On Positivity of Polynomials: The Dilation Integral Method

The focal point of this paper is the well known problem of polynomial positivity over a given domain. More specifically, we consider a multivariate polynomial
f
(
x
) with parameter vector
x
restricted to a hypercube
X
sub
R

n
. The objective is to determine if
f
(
x
) > 0 for all
x
isin
X
. Motivated by NP-Hardness considerations, we introduce the so-called dilation integral method. Using this method, a ldquosofteningrdquo of this problem is described. That is, rather than insisting that
f
(
x
) be positive for all
x
isin
X
, we consider the notions of practical positivity and practical non-positivity. As explained in the paper, these notions involve the calculation of a quantity epsiv > 0 which serves as an upper bound on the percentage volume of violation in parameter space where
f
(
x
) les 0 . Whereas checking the polynomial positivity requirement may be computationally prohibitive, using our epsiv-softening and associated dilation integrals, computations are typically straightforward. One highlight of this paper is that we obtain a sequence of upper bounds epsiv
k
which are shown to be ldquosharprdquo in the sense that they converge to zero whenever the positivity requirement is satisfied. Since for fixed
n
, computational difficulties generally increase with
k
, this paper also focuses on results which reduce the size of the required
k
in order to achieve an acceptable percentage volume certification level. For large classes of problems, as the dimension of parameter space
n
grows, the required
k
value for acceptable percentage volume violation may be quite low. In fact, it is often the case that low volumes of violation can be achieved with values as low as
k
=2.