Convexifying Positive Polynomials and Sums of Squares Approximation
نویسندگان
چکیده
منابع مشابه
Convexifying 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...
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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...
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If a real polynomial f can be written as a sum of squares of real polynomials, then clearly f is nonnegative on R, and an explicit expression of f as a sum of squares is a certificate of positivity for f . This idea, and generalizations of it, underlie a large body of theoretical and computational results concerning positive polynomials and sums of squares. In this survey article, we review the...
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Acknowledgements A lot of people helped me during the work on this thesis. First of all, I am greatly indebted to Alexander Prestel, Claus Scheiderer and Markus Schweighofer. I benefitted a lot from their constant logistic and mathematical support. I also want to thank Robert Denk, David Grimm and Daniel Plaumann for many interesting discussions on the topic of this work. In 2007, I enjoyed a s...
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ژورنال
عنوان ژورنال: SIAM Journal on Optimization
سال: 2015
ISSN: 1052-6234,1095-7189
DOI: 10.1137/140958165