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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)

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)

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)

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)

In recent years, much work has been devoted to a systematic study of polynomial identities certifying strict or non-strict posi-tivity of a polynomial f on a basic closed set K ⊂ R n. The interest in such identities originates not least from their importance in polynomial optimization. The majority of the important results requires the archimedean… (More)

A linear matrix inequality (LMI) is a condition stating that a symmetric matrix whose entries are affine linear combinations of variables is positive semidefinite. Motivated by the fact that diagonal LMIs define poly-hedra, the solution set of an LMI is called a spectrahedron. Linear images of spectrahedra are called semidefinitely representable sets. Part… (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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