Параметры публикации

Тип публикации: 



On Positivity of Polynomials: The Dilation Integral Method

Наименование конференции: 

  • IEEE Transactions on Automatic Control


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Год издания: 



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 XМRn. The objective is to determine if f(x)>0 for all xОX. Motivated by NP-Hardness considerations, we introduce the so-called dilation integral method. Using this method, a «softening» of this problem is described. That is, rather than insisting that f(x) be positive for all xОX, we consider the notions of practical positivity and practical non-positivity. As explained in the paper, these notions involve the calculation of a quantity e > 0 which serves as an upper bound on the percentage volume of violation in parameter space where f(x) Ј 0: Whereas checking the polynomial positivity requirement may be computationally prohibitive, using our І- softening and associated dilation integrals, computations are typically straightforward. One highlight of this paper is that we obtain a sequence of upper bounds e k which are shown to be «sharp» in the sense that they converge to zero whenever the positivity requirement is satisfied. Since for fixed n computational dificulties 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.

Библиографическая ссылка: 

Barmish B. R., Щербаков П.С., Ross S. R., Dabbene F. On Positivity of Polynomials: The Dilation Integral Method / . -: -, 2009. С. -.