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Representations of positive polynomials on noncompact semialgebraic sets via KKT ideals
, 2006
"... This paper studies the representation of a positive polynomial f(x) on a noncompact semialgebraic set S = {x ∈ R n: g1(x) ≥ 0, · · · , gs(x) ≥ 0} modulo its KKT (KarushKuhnTucker) ideal. Under the assumption that the minimum value of f(x) on S is attained at some KKT point, we show that f(x) ..."
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Cited by 19 (4 self)
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This paper studies the representation of a positive polynomial f(x) on a noncompact semialgebraic set S = {x ∈ R n: g1(x) ≥ 0, · · · , gs(x) ≥ 0} modulo its KKT (KarushKuhnTucker) ideal. Under the assumption that the minimum value of f(x) on S is attained at some KKT point, we show that f(x) can be represented as sum of squares (SOS) of polynomials modulo the KKT ideal if f(x)> 0 on S; furthermore, when the KKT ideal is radical, we have that f(x) can be represented as sum of squares (SOS) of polynomials modulo the KKT ideal if f(x) ≥ 0 on S. This is a generalization of results in [18], which discuss the SOS representations of nonnegative polynomials over gradient ideals. Key words: Polynomials, semialgebraic set, sum of squares (SOS), KarushKuhnTucker (KKT) system, KKT ideal. 1
POLYNOMIALS NONNEGATIVE ON A STRIP
"... Abstract. We prove that if f(x, y) is a polynomial with real coefficients which is nonnegative on the strip [0, 1] × R, then f(x, y) has a presentation of the form k∑ f(x, y) = gi(x, y) 2 ℓ∑ + hj(x, y) 2 x(1 − x), i=1 j=1 where the gi(x, y) and hj(x, y) are polynomials with real coefficients. 1. ..."
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Cited by 9 (2 self)
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Abstract. We prove that if f(x, y) is a polynomial with real coefficients which is nonnegative on the strip [0, 1] × R, then f(x, y) has a presentation of the form k∑ f(x, y) = gi(x, y) 2 ℓ∑ + hj(x, y) 2 x(1 − x), i=1 j=1 where the gi(x, y) and hj(x, y) are polynomials with real coefficients. 1.
Cylinders with compact crosssection and the strip conjecture
, 2008
"... The proof of the strip conjecture given just recently in [7] (see Theorem 3.1 below for a statement of the result) is best understood as a refinement of a proof of a general result for cylinders with compact crosssection (see Theorem 2.2 below). We elaborate on this statement. We give a detailed pr ..."
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Cited by 3 (1 self)
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The proof of the strip conjecture given just recently in [7] (see Theorem 3.1 below for a statement of the result) is best understood as a refinement of a proof of a general result for cylinders with compact crosssection (see Theorem 2.2 below). We elaborate on this statement. We give a detailed proof of Theorem 2.2 and we
Minimizing Polynomials Over Semialgebraic Sets
, 2005
"... This paper concerns a method for finding the minimum of a polynomial on a semialgebraic set, i.e., a set in R m defined by finitely many polynomial equations and inequalities, using the KarushKuhnTucker (KKT) system and sum of squares (SOS) relaxations. This generalizes results in the recent paper ..."
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Cited by 1 (0 self)
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This paper concerns a method for finding the minimum of a polynomial on a semialgebraic set, i.e., a set in R m defined by finitely many polynomial equations and inequalities, using the KarushKuhnTucker (KKT) system and sum of squares (SOS) relaxations. This generalizes results in the recent paper [15], which considers minimizing polynomials on algebraic sets, i.e., sets in R m defined by finitely many polynomial equations. Most of the theorems and conclusions in [15] generalize to semialgebraic sets, even in the case where the semialgebraic set is not compact. We discuss the method in some special cases, namely, when the semialgebraic set is contained in the nonnegative orthant R n + or in box constraints [a, b]n. These constraints make the computations more efficient.
Positive Hybrid RealTrigonometric Polynomials and Applications to Adjustable Filter Design and Absolute Stability Analysis
"... Abstract The present paper points out that a class of positive polynomials deserves special attention due to several interesting applications in signal processing, system analysis and control. We consider positive hybrid polynomials with two variables, one real, the other complex belonging to the u ..."
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Abstract The present paper points out that a class of positive polynomials deserves special attention due to several interesting applications in signal processing, system analysis and control. We consider positive hybrid polynomials with two variables, one real, the other complex belonging to the unit circle. We present several theoretical results regarding the sumofsquares representations of such polynomials, treating the cases where positivity occurs globally or on domains. We also give a specific Bounded Real Lemma. All the characterizations of positive hybrid polynomials are expressed in terms of positive semidefinite matrices and can be extended to polynomials with more than two variables. On the applicative side, we show how several problems are numerically tractable via semidefinite programming (SDP) algorithms. The first problem is the minimax design of adjustable FIR filters, using a modified Farrow structure. We discuss linearphase and approximately linearphase designs. The second is the absolute stability of time delay feedback systems with unknown delay, for which we treat the cases of bounded and unbounded delay. Finally, we discuss the application of our methods to checking the stability of parameterdependent systems. The design procedures are illustrated with numerical examples.