Learn More
Let S = {x ∈ R n | g 1 (x) ≥ 0,. .. , gm(x) ≥ 0} be a basic closed semialgebraic set defined by real polynomials g i. Putinar's Positivstellensatz says that, under a certain condition stronger than compactness of S, every real polynomial f positive on S posesses a representation f = P m i=0 σ i g i where g 0 := 1 and each σ i is a sum of squares of(More)
A basic closed semialgebraic subset S of R n is defined by simultaneous polynomial inequalities g 1 ≥ 0,. .. , gm ≥ 0. We give a short introduction to Lasserre's method for minimizing a polynomial f on a compact set S of this kind. It consists of successively solving tighter and tighter convex relaxations of this problem which can be formulated as(More)
Farkas' lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly in-feasible linear matrix inequalities. We provide nonlinear algebraic certificates for all infeasible linear matrix inequalities in the(More)
We consider the problem of computing the global infimum of a real polynomial f on R n. Every global minimizer of f lies on its gradient variety, i.e., the algebraic subset of R n where the gradient of f vanishes. If f attains a minimum on R n , it is therefore equivalent to look for the greatest lower bound of f on its gradient variety. Nie, Demmel and(More)
Let A be a commutative R–algebra of finite transcendence degree d ∈ N. We investigate the relationship between the subring of (geometrically) bounded elements H(A) := {a ∈ A | ∃ν ∈ N : |a| ≤ ν on Sper A} and the subring of arithmetically bounded elements H (A) := {a ∈ A | ∃ν ∈ N : ν + a and ν − a are sums of squares in A}. Obviously, H (A) ⊆ H(A). In 1991,(More)