Автор(ы): Barmish B. R. (?)Щербаков П. С. (ИПУ РАН, Лаборатория 07)Ross S. R. (?)Dabbene F. (?)Автор(ов): 4 Параметры публикацииТип публикации: ДокладНазвание: On Positivity of Polynomials: The Dilation Integral MethodНаименование конференции: IEEE Transactions on Automatic ControlГород: -Издательство: -Год издания: 2009Страницы: - АннотацияThe 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. -.