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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)

Schmüdgen's Positivstellensatz roughly states that a polynomial f positive on a compact basic closed semialgebraic subset S of R n can be written as a sum of polynomials which are nonnegative on S for certain obvious reasons. However, in general, you have to allow the degree of the summands to exceed largely the degree of f. Phenomena of this type are one… (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)

A linear matrix inequality (LMI) is a condition stating that a symmetric matrix whose entries are affine-linear combinations of variable,; is positive semidefinite. :dotiv11ted by the fact that diagonal LMIs define polyhedra, the solution set of an LMI is called a spectrahedron. Linear images of spectrahedra are called semidefinitely representable sets.… (More)

A <i>semialgebraic</i> set is a set described by a boolean combination of real polynomial inequalities in several variables. A <i>linear matrix inequality</i> (LMI) is a condition expressing that a symmetric matrix whose entries are affine-linear combinations of variables is positive semidefinite. We call solution sets of LMIs <i>spectrahedra</i> and their… (More)

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