Polynomials Positive on Unbounded Rectangles
نویسندگان
چکیده
Given a K ⊆ R determined by a finite set of {g1 ≥ 0, . . . , gk ≥ 0}, we want to characterize a polynomial f which is positive (or non-negative) on K in terms of sums of squares and the polynomials gi used to describe K. Such a representation of f is an immediate witness to the positivity condition. Theorems about the existence of such representations also have various applications, notably in problems of optimizing polynomial functions on semialgebraic sets. In case K is compact, has proved that any polynomial which is positive on K is in the preorder generated by the gi’s, i.e., the set of finite sums of elements of the form seg 1 . . . g ek k , ei ∈ {0, 1}, where each se is a sum of squares of polynomials. has proved that, under certain conditions, the preorder can be replaced by the , which is the set of sums {s0 + s1g1 + · · ·+ skgk}, where each si is a . Using this result, has developed algorithms for finding the minimum of a polynomial on such compact K, which transforms this into a semidefinite programming problem. What happens whenK is not compact? has shown that ifK is not compact and dimK ≥ 3, then Schmüdgen’s characterization can never hold, regardless
منابع مشابه
Numerical solution of Fredholm integral-differential equations on unbounded domain
In this study, a new and efficient approach is presented for numerical solution of Fredholm integro-differential equations (FIDEs) of the second kind on unbounded domain with degenerate kernel based on operational matrices with respect to generalized Laguerre polynomials(GLPs). Properties of these polynomials and operational matrices of integration, differentiation are introduced and are ultili...
متن کاملPositive polynomials on fibre products
Recent investigations in optimization theory concerning the structure of positive polynomials with a sparsity pattern are interpreted in the more invariant language of (iterated) fibre products of real algebraic varieties. This opens the perspective of treating on a unifying basis the cases of positivity on unbounded supports, on non-semialgebraic supports, or of polynomials depending on counta...
متن کاملApproximation of positive polynomials by sums
This overview is intended to provide an “ atlas ” of what is known about approximations of the cone of positive polynomials (on a semialgebraic set KS) by various preorderings (or the corresponding module versions). These approximations depend on the description S of KS, the dimension of the semi-algebraic set KS, intrinsic geometric properties of KS (e.g. compact or unbounded), and special pro...
متن کاملOn Long Time Integration of the Heat Equation
We construct space-time Petrov–Galerkin discretizations of the heat equation on an unbounded temporal interval, either right-unbounded or left-unbounded. The discrete trial and test spaces are defined using Laguerre polynomials in time and are shown to satisfy the discrete inf-sup condition. Numerical examples are provided.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2004