Approximation of positive polynomials by sums
نویسنده
چکیده
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 properties of KS (symmetry, sparse representation)
منابع مشابه
Workshop Optimization and Control . IMA Minneapolis , January 16 – 20 2007
Approximation of positive polynomials by sums of squares.
متن کاملSums of Squares, Moment Matrices and Optimization over Polynomials
We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equations and inequalities, which is NP-hard in general. Hierarchies of semidefinite relaxations have been proposed in the literature, involving positive semidefinite moment matrices and the dual theory of sums of squares of polynomials. We present these hierarchies of approximations and their main...
متن کاملConvexifying Positive Polynomials and Sums of Squares Approximation
We show that if a polynomial f ∈ R[x1, . . . , xn] is nonnegative on a closed basic semialgebraic set X = {x ∈ Rn : g1(x) ≥ 0, . . . , gr(x) ≥ 0}, where g1, . . . , gr ∈ R[x1, . . . , xn], then f can be approximated uniformly on compact sets by polynomials of the form σ0 + φ(g1)g1 + · · ·+ φ(gr)gr, where σ0 ∈ R[x1, . . . , xn] and φ ∈ R[t] are sums of squares of polynomials. In particular, if X...
متن کاملNumerical solution of the spread of infectious diseases mathematical model based on shifted Bernstein polynomials
The Volterra delay integral equations have numerous applications in various branches of science, including biology, ecology, physics and modeling of engineering and natural sciences. In many cases, it is difficult to obtain analytical solutions of these equations. So, numerical methods as an efficient approximation method for solving Volterra delay integral equations are of interest to many res...
متن کاملExtremal Positive Trigonometric Polynomials
There are various reasons for the interest in the problem of constructing nonnegative trigonometric polynomials. Among them are: Cesàro means and Gibbs’ phenomenon of the the Fourier series, approximation theory, univalent functions and polynomials, positive Jacobi polynomial sums, orthogonal polynomials on the unit circle, zero-free regions for the Riemann zeta-function, just to mention a few....
متن کامل